► 


VITAL  STATISTICS. 

BY 

f  E.  B.  ELLIOTT, 

|  '■’S'-.  OF  BOSTON. 


From  the  Proceedings  of  the  American  Association  for  the  Advancement  of 

Science. 


* 


50 


A.  MATHEMATICS  AND  PHYSICS. 


u 


VITAL  STATISTICS. 

A.  Tables  of  Prussian  Mortality ,  interpolated  for  Annual  Intervals 
of  Age ;  accompanied  with  Formulce  and  Process  for  Construction. 

B.  Discussion  of  Certain  Methods  for  converting  Ratios  of  Deaths  to 
Population ,  within  given  Intervals  of  A ge,  into  the  Logarithm  of 
the  Probability  that  one  living  at  the  Earlier  Age  will  attain  the 
Later  ;  with  Illustrations  from  English  and  Prussian  Data. 

C.  Process  for  deducing  accurate  Average  Duration  of  Life ,  present 
Value  of  Life- Annuities ,  and  other  useful  Tables  involving  Life- 
Contingencies,  from  Returns  of  Population  and  Deaths,  without  the 
Intervention  of  a  General  Interpolation. 

The  mortality  and  accompanying  tables,  to  which  the  attention  of 
the  Association  is  called,  comprise  portions  of  a  series  of  tables  that 
have  been  and  are  being  prepared,  for  the  New  England  Mutual  Life 
Insurance^  Company  of  Boston,  from  official  returns  of  the  British, 
Swedish,  Prussian,  and  Belgian  governments,  and  from  such  reliable 
American  statistics  as  are  obtainable. 

In  several  of  the  United  States  of  America  the  decennial  enumer¬ 
ation  of  the  numbers  and  ages  of  the  living  effected  for  the  General 
Government  have  been  quite  accurate  and  reliable,  while  the  only 
official  mortality  returns  (viz.  those  ordered  in  connection  with  the 
last  census,  1850)  are  inaccurate  and  deficient.  In  Massachusetts, 
since  its  Registration  Act  of  1849,  certain  districts  have  furnished 
valuable  and  satisfactory  information  respecting  the  numbers  and  ages 
of  the  dying  ;  but  from  the  published  abstracts  it  has  been  impossible 
to  separate  imperfect  from  reliable  data.  In  the  yet  unpublished 
abstracts  of  the  returns  for  1855  an  improvement  is  being  effected, 
under  the  direction  of  the  present  Secretary  of  State,  which,  although 
augmenting  somewhat  the  expense,  will  afford  fit  material  for  the 
construction  of  a  Life -Table  that  shall  satisfactorily  represent  the  rates 
of  mortality  prevailing  among  the  inhabitants  of  the  larger  part  of  the 
Commonwealth. 

The  leading  paper  (A)  presents  a  new  Life-Table,  complete  for 
annual  intervals  of  age,  and  calculated  from  over  a  million  (1,197,407) 


V\A  I. 

^  r'  -‘  "  ?’  uV^ne-nnajti  c  *5 

MATHEMATICS.  51 

_  A  * 

> 

of  observations  regarding  the  ages  of  the  dying,  in  a  population  of 
fifteen  millions  (14,928,501),  and  in  a  community  where  observations 
a—  on  vital  statistics,  for  many  years,  are  believed  to  have  been  made  with 
care  and  accuracy.  It  adds  one  to  the  very  limited  list  of  National 
Life-Tables. 

The  remaining  papers  (B  and  C)  are  devoted  to  the  discussion  of 
certain  methods  for  converting  rates  of  mortality  for  different  inter¬ 
vals  of  age  into  probability  of  living  ;  and  to  the  presentation  of 
abridged  methods  for  calculating,  at  certain  ages,  accurate  tables  of 
-  practical  value,  involving  life-contingencies,  accompanied  with  simple 
rules  for  determining  any  required  value  intermediate. 


A.  Tables  of  Prussian  Mortality,  interpolated  for  Annual  Inter¬ 
vals  of  Age  ;  accompanied  with  Formulae  and  Process  for  Con¬ 
struction. 

The  data  from  which  the  following  tables  have  been  calculated 
were  obtained  from  documents  sent  by  Mr.  Hoffman  of  Berlin  to  the 
English  Ministry  of  Foreign  Affairs,  and  published  in  the  Sixth  Annual 
Report  of  the  Registrar-General  in  England. 

Population  of  Prussia,  Civil  and  Military  (exclusive  of  Neuf- 

chatel).* 

At  the  end  of  the  year  1834,  13,509,927. 

46  44  1837,  14,098,125. 

44  44  1840,  14,928,501. 

The  documents  above  mentioned  give  no  statistics  of  immigration  or 

emigration. 

The  increase  of  population  during  the  three  years  1838,  ’39,  ’40, 
was  830,376. 

The  excess  of  births  over  deaths  during  the  same  three  years  was 

486,937. 

Leaving  343,439,  which  is  41.36  per  cent  of  the  total  increase  of 
population,  unaccounted  for  by  excess  of  births  over  deaths. 


*  “  The  population  of  Neufchatel,  not  included  in  the  above,  was  59,448  in  1837, 
52,223  in  1825” 

^2.  £$4  2- 


52 


A.  MATHEMATICS  AND  PHYSICS, 


Population  of  Prussia  at  the  End  of  the  Year  1840,  classed  accord¬ 
ing  to  Age  and  Sex. 


Ages. 

Males. 

Females. 

Males. 

Females. 

Ages. 

0-  5 

1,134,413 

1,114,871 

) 

5-  7 

370,740 

336,429 

( 

>  2,603,699 

2,550,022 

0-14 

7-14 

1,098,546 

1,068,722 

\ 

14-16 

344,179 

331,039 

14-16 

16-20 

586,059 

20-25 

692,704 

25-32 

777,183 

-3,238,434 

3,253,643 

16-45 

32-39 

646,122 

39-45 

536,366 

45-60 

816,726 

881,280 

45-60 

60  and  upwards, 

445,544 

463,935 

.  60  and  upwards. 

All  Ages, 

7,448,582 

7,479,919 

Assuming  the  distribution  of  the  (3,253,643)  females  for  the  several 
intervals  between  the  ages  16  and  45  to  be  proportioned  to  the  distri¬ 
bution  of  (3,238,434)  the  corresponding  number  of  males,  we  have 


Ages. 

Females. 

16-20 

588,812 

20-25 

695,957 

25-32 

780,833 

32-39 

649,156 

39-45 

538,885 

Total,  16-45 

3,253,643 

Hence  the  following 

Numbers  and  Ages  of  the  Population  of  Prussia  at  the  End  of  the 

Year  1840. 


Ages. 

Persons. 

0-  5 

5-  7 

7-14 

14-16 

16-20 

20  -  25 

25-32 

32-39 

39-45 

45-60 

60  and  upwards, 

2,249.284 

737,169 

2,167,268 

675,218 
1,174,871 
1,388,661 
-  1,558,016 
1,295.278 
1,075,251 
1,698,006 

909,479 

All  Ages, 

1 

14,928,501 

We  wish  to  distribute  the  population  from  ages  25  to  45,  from  45  to 


MATHEMATICS. 


53 


60,  and  from  60  upwards  in  quinquennial  or  decennial  periods,  to  corre¬ 
spond  with  the  ages  of  the  dying  as  presented  in  the  mortality  returns. 

We  first  determine  the  quinquennial  distribution  between  ages  25 
and  45. 

Let  Px/y  represent  the  population  between  ages  x  and  y,  or  the 
numbers  living  under  age  y,  less  the  numbers  living  under  age  x. 


P 16/20 

=  1,174,871 

P 16/25 

=  2,563,532 

P 16/32 

=  4,121,548 

P 16/39 

=  5,416,826 

P 16/45 

=  6,492,077 

P 16/60 

=  8,190,083 

Let  a  =  20,  b  =  25,  c 
Let  a  =  25,  b  =  32,  c 

Let  a  —  32,  b  =  39,  c 
Let  a  =  39,  b  =  45,  c 


Assume  P16/,  =  Pl6/a 

+  P 16/6 

4“  P 1 6/c 

32  ;  then  will  Pi6/3o  = 

gq  #  ( then  will  P16/30  — 

’ 1  and  P16/35  = 

then  will  P16/35  = 

and  P  /40  = 

60  ;  then  will  P16/40  = 


x —b . x - c 
a-b  . a- c 
x  —  a . x-c 
b-a . b -c 
x-a . x-b 
c-a . c-b 
:  3,722,366. 
3,703,211, 
4,708,839. 
4,682,050, 
:  5,598,277. 
:  5,611,751. 


} 

} 

i 


Taking  the  arithmetical  mean  of  the  above  duplicate  results,  we 
have 


Pl6l30  =  3,712,789 

Hence  P25,30  =  1,149,257 

P 16/35  =  4,695,445 

P 30/35  =  982,656 

■P.6,40  =  5,605,014 

Pas/40  =  909,569 

P 40/15  =  887,063 

which  results  cannot  vary  materially  from  the  actual  distribution. 

The  following  Table  gives  the  distribution  of  the  population  of 
Prussia  between  ages  45  and  60  (1,698,006)  ;  and  of  the  population 
from  age  60  upwards,  according  to  the  corresponding  proportional  dis¬ 
tribution  of  the  numbers  of  the  population  of  the  Northwestern  Division 
of  England  (the  Eighth  of  the  eleven  Districts  into  which  England  and 
Wales  are  divided  in  the  Reports  of  the  Registrar-General). 


Ages. 

Distribution  of  Prussian  Population  over  age  45. 

45-55 

1,257,322 

55-60 

440,684 

60-65 

353,657 

65-75 

398,925 

75-85 

137,188 

85  -  95 

18,638 

95  and  over, 

1,026 

1  1 

45  and  over, 

2,607,485 

5  * 


54 


A.  MATHEMATICS  AND  PHYSICS. 


Numbers  and  Ages  of  the  Population  of  the  Northwestern  Di¬ 
vision  (Eng.)  in  1841,  according  to  which  the  above  Distribution 
was  made.  —  (9th  Rep.  Reg.-Gen.) 


Ages. 

Northwestern  Division  (Eng.),  1841. 
Numbers  and  Ages  of  the  Living  above  Age  45. 

45-55 

151,064 

55  -  60 

52,947 

60  -  65 

39,656 

65  -  75 

44,732 

75-85 

15,383 

85-95 

2,095 

95  and  over, 

115 

1 

45  and  over, 

305,992 

The  numbers  living  above  age  15  in  the  Northwestern  Division 
(England)  were  grouped,  with  reference  to  age,  only  in  decennial 
classes. 

By  assuming  the  algebraic  equation, 


55  lx 


P55 


55/45 


x — 55  .  x  —  65  .  *  —  75 
45  — 55 . 45  — 65 . 45  —  75 


x — 45.07  —  55.0:  —  75 
+  i  si65  ■  65  —  45  .  65  —  55 . 65  —  75 


+  P: 


07  —  45 . 07  —  55 . 07 — 65 


55/75 


75  —  45 . 75  —  55 . 75  —  65 
a  close  approximation  to  the  probable  number  of  persons  living  be¬ 
tween  ages  55  and  60  (52,947)  and  between  ages  60  and  65  (39,656) 
resulted. 

P55/x  represents  the  number  of  persons  reported  living  under  age  07, 
less  the  number  living  under  age  55. 

We  remark  that  P55/ 45  is  essentially  negative. 

The  population  of  the  Division,  as  returned  for  the  night  of  June 
6-7,  1841,  was  two  millions  (2,098,820),  being  one  eighth  of  the 
entire  population  of  England  and  Wales  (15,914,148)  at  that  date. 
The  counties  of  Cheshire  and  Lancashire  constitute  this  Division. 
The  latter  county  includes  the  densely  populous  and  unhealthy  district^ 
of  Liverpool .  /fyl 

The  ratios  of  deaths  to  population  for  the  intervals  from  age  45  to 
60,  and  from  age  60  upwards,  more  closely  approximated  the  corre¬ 
sponding  ratios  for  Prussia,  than  did  those  of  any  other  large  commu¬ 
nity  concerning  which  reliable  population  and  mortality  statistics  were 
to  be  obtained. 


MATHEMATICS. 


55 


Table  comparing  Ratios  of  the  Annual  Number  of  Deaths  to  the 
Numbers  living  in  certain  Communities  from  Age  45  to  60,  and 
from  Age  60  to  extreme  Old  Age. 


Ages 

45  -  60. 

Ages 

60  and  upwards. 

Prussia. 

Deaths,  1839,  ’40,  ’41, 1 

.02 4^? 

* 0 
.08 9- 

Population,  1840,  J 

Northwestern  Division  [England). 

Deaths,  seven  years,  1838  -  44,  \ 

Population,  middle,  1841,  1 

.023 

.079+ 

Sweden. 

Deaths,  twenty  years,  1821  -40,  ) 

Population,  mean  of  1820, ’30, ’40,  J 

.021 

.079— 

Belgium. 

Deaths,  nine  years,  1842-50,  ) 

Population,  October  15,  1856,  J  ’ 

.020 

.073 

Enqland  and  Wales. 

1841,  . 

.019 

.069 

A  comparison  of  the  distribution  of  the  numbers  of  the  living  in 
Prussia  in  these  intervals  of  age  according  to  that  of  the  North¬ 
western  Division  (.Eng.),  with  a  distribution  of  the  same  numbers 
according  to  the  mean  of  the  corresponding  distribution  of  equal 
numbers  of  the  populations  of  England  in  1841  and  of  Belgium  in 
1846,  would  give  the  following  results. 

Distribution  of  the  Population  of  Prussia  according  to 


I 

Ages. 

The  Mean  of  Equal  Numbers 

The  Northwestern  Division 

in  England  and  Belgium. 

(England). 

45-55 

1,285,567 

1,257,322 

55-60 

412,439 

440,684 

45-60 

1,698,006 

1,698,006 

60-65 

330,425 

353,657 

65-75 

398,646 

398,925 

75-85 

155,077 

137,188 

85  and  over, 

25,331 

19,709 

60  and  over, 

909,479 

909,479 

45  and  over, 

2,607,4^85  v 

2,607,485 

The  distribution  according  to  the  English  and  Belgian  facts  would 
give  larger  numbers  after  about  age  75,  in  the  resulting  Life-Table. 

The  distribution  according  to  that  of  the  Northwestern  Division 
was  adopted  as  the  best  representation  of  the  probable  corresponding 
distribution  of  the  population  of  Prussia,  within  the  intervals  of  age 
above  mentioned.  Hence  the  following  Table. 


56 


A.  MATHEMATICS  AND  PHYSICS, 


Deaths,  Population,  Mortality,  and  Logarithms  of  the  Probability 
of  Living  in  Prussia. 

The  Numbers  of  the  Living  between  ages  45  and  60,  and  from  60  to  extreme  old  age 
are  distributed  according  to  corresponding  proportional  distributions  of  the  numbers  of 
the  population  of  the  Counties  of  Cheshire  and  Lancashire  ( Northwestern  Division ), 
in  England ,  in  1841. 


Deaths. 


Aggregate 
Numbers 
and  Ages  of 
the  Dying 
during  the 
Three  Years 
1839,  ’40,  ’41 


Population. 


Numbers 
and  Ages  of  the 
Living  at  the 
End  of  the 
Year  1840. 


Ratios  of  the  Average 
Annual  Numbers  of 
the  Dying  during  the 
Three  Years  1839,  ’40. 
’41,  to  the  Numbers 
of  the  Living  com¬ 
puted  with  reference 
to  the  Middle  of  1840. 


Mortality. 


Logarithms  of  Probabil¬ 
ities  of  Surviving 
each  Interval. 


Duplicate 
Values,  each 
deduced  from 
two  consecutive 
Ratios  in  the 
Column  of 
Mortality. 


Values 
derived  from 
Comparison  of 
the  Duplicates. 


Ages. 

x>  y- 


D0,V  ®0,x 


P  —P 

0/y  X  0/x 


D 


^P  o/y  ^Po/x 


M 


XPx 


0-1 

1-3 

3-5 

5-7 

7-14 

14-20 

20-25 

25-30 

30-35 

35-40 

40-45 

45-55 

55-60 

60-65 

65-75 

75-85 
85  and 
upwards 

Total, 


310,527 

162,356 

62,734 

33,272 

27,156 


34,585 

36,849 

31,594 

35,579 

38,094 

78,503 

46,704 

58,576 

107,653 

61,697 

15,572 


2,249,284 

737,169 

2,167,268 

1,850,089 

1,388,661 

1,149,257 

982.656 
909,569 
887,063 

1,257,322 

440,684 

353.657 
398,925 

137,188 

19,709 


.0802238 

.0152056 

.0077790 

.0062978 

.0089397 

.0096939 

.0108317 

.0131780 

.0144675 

.0210345 

.0357042 

.0557995 

.0909134 

.1515098 

.2661784 


1,197,407 


14,928,501 


—  .013106 

—  .013201 

—  .023480 

—  .023628 

—  .016399 

—  .016432 

—  .019433 

—  .019418 

—  .021057 

—  .021059 

—  .023533 

—  .023542 

—  .028646 

—  .028630 

—  .031434 

—  .031464 

—  .092155 

—  .092527 

—  .077947 

—  .078021 

—  .122543 

—  .121891 

—  .424020 

—  .408584 

—  .716433 

—  .730677 


—  .013155 

—  .023557 

—  .016416 

—  .019425 

—  .021058 

—  .023537 

—  .028637 

—  .031449 

—  .092322 

—  .077981 

—  .122189 

—  .415608 

—  .722021 


1492850 1  —  Population  of  Prussia,  as  returned  for  the  end  of  the  year  1840. 


14770727  —  Population  of  Prussia,  estimated  for  the  middle  of  the  year  1840, 
from  the  numbers  returned  as  living  at  the  end  of  1834, 1837, 
and  1840. 


MATHEMATICS. 


57 


It  will  be  observed  that  the  values  derived  from  comparison  of  the 
duplicate  logarithms,  and  which  have  been  adopted  in  constructing 
the  Interpolated  and  other  Tables,  are  not  in  all  cases  arithmetical 
means.  The  difference  is  of  little  moment,  but  there  is  no  sufficient 
reason  for  preferring  the  former. 

Logarithms  of  the  Probability  of  Surviving, 

Computed  from  the  Returns  of  the  Numbers  of  the  Living  under  Age  5 ;  and  of  the  Numbers 
of  the  Dging  annually  under  1  Year  of  Age,  over  1  and  under  3,  over  3  and  under  5. 


Ages. 

^Px,y: 

0-1 

—  .082920 

1-3 

—  .051670 

3-5 

—  .022522 

The  successive  addition  of  the  logarithms  of  the  probabilities  of 
surviving  the  consecutive  intervals  of  age  to  5.001688,  the  logarithm 
assumed  for  the  proportional  numbers  born  alive,  gives  the  following 

Table  of  the  Logarithms  of  the  Proportions,  and  the  Proportions 
of  Persons  born  alive  and  surviving  certain  Ages  in  Prussia,  ac¬ 
cording  TO  THE  CALCULATED  LAW  OF  MORTALITY. 

Deaths ,  1839,  ’40,  ’41. 

Population  computed  with  reference  to  middle  of  1 840. 

Distribution  of  Population  above  Age  45,  Northwestern  Division  {Eng.). 


1 

Survivors. 

Age. 

Logarithms. 

Persons. 

X  L 

L 

X 

X 

1 

0 

5.001688 

100,389 

1 

4.918768 

82,941 

3 

4.867098 

73,637 

5 

4.844576 

69,916 

14 

4.807864 

64,249 

25 

4.772023 

59,159 

35 

4.727428 

53,386 

45 

4.667342 

46,488 

55 

4.575020 

37,585 

65 

4.374850 

23,706 

75 

3.959242 

9,104.2 

85 

3.237221 

1,726.7 

j  95 

1.982879 

96.1 

105 

1.803755 

.636 

58 


A.  MATHEMATICS  AND  PHYSICS. 


The  values  opposite  ages  95  and  105  were  computed  from  the 
logarithms  of  the  numbers  surviving  at  ages  65,  75,  and  85,  by  the 
exponential  formula, 

X  Lx  =  $x  =  $65  +  ($75 - $65 )  Jt5~-65_  j  5 

in  which 


0 


75  —  65 


(or  q'°)  = 


$85 - $75 

$75 -  $65* 


These  values  were  adopted  as  bases  for  the  construction  of  the 
accompanying  Life-Table  interpolated  for  annual  intervals  of  age ; 
and  also  for  computing  by  abridged  methods  certain  practical  life- 
contingency  tables. 

Before  presenting  this  table  and  these  methods,  we  will  state  some 
of  the  principles  which  underlie,  and  indicate  the  process  by  which 
ratios  of  the  numbers  of  the  dying  to  the  numbers  of  the  living,  during 
the  several  intervals  of  age,  have  been  converted  into  logarithms  of 
the  probabilities  that  one  living  at  the  earlier  age  will  attain  the  later. 

Whenever,  in  any  community,  the  intensity  of  mortality  at  each 
age,  or  the  ratio  of  the  numbers  momentarily  dying  during  each 
minute  interval  of  age  to  the  numbers  then  living  within  the  same 
interval,  has  been  constant  for  a  period  of  time  equal  to  the  difference 
between  the  specified  age  and  the  extreme  of  old  age,  an  invariable 
law  of  mortality  is  said  to  prevail  in  that  community. 

The  law  of  human  mortality  is  seldom  strictly  invariable.  It  fluc¬ 
tuates  within  certain  limits ,  not  only  with  different  communities  and 
localities,  but  in  the  same  community  during  successive  periods,  and 
in  the  same  localities.  The  habits,  occupations,  and  social  condition 
of  the  members  of  the  community  remaining  unchanged,  the  larger 
their  numbers  the  narrower  these  limits.  It  is  within  the  province 
of  the  vital  statistician  to  determine,  not  merely  an  average  of  the 
rates  of  mortality  prevailing  in  a  community,  but  also  the  sensible 
limits  within  which  the  rates  fluctuate. 

Our  present  inquiries  have  reference  to  the  determination  of  a  law 
of  mortality  which  shall  satisfactorily  represent  the  average  of  the 
rates  prevailing  among  the  inhabitants  of  a  populous  state,  with  fixed 
geographical  boundaries  ;  and  in  which  the  numbers  of  the  inhab¬ 
itants  vary  with  births  and  with  deaths,  with  immigration  and  with 
emigration. 


MATHEMATICS. 


59 


If,  in  a  large  community,  varying  with  births,  deaths,  and  migra¬ 
tions,  but  in  which  the  numbers  of  the  population  have  not  been 
subject  to  sudden  and  irregular  change,  the  number  of  the  dying 
during  a  given  year  or  period  of  time  between  ages  not  very  remote 
be  divided  by  the  number  of  the  living  between  the  same  ages  at 
the  middle  of  that  period  of  time,  the  quotient  resulting  from  the 
division  has  generally  been  assumed  closely  to  approximate  the  quo¬ 
tient  that  would  have  resulted  had  the  numbers  of  the  population 
within  the  limits  of  these  ages  been  stationary ;  that  is,  assuming 
an  invariable  law  of  mortality,  had  the  numbers  of  the  population 
for  each  minute  interval  of  age  within  the  limits  of  these  ages  re¬ 
mained  constant  during  that  period,  and  unaffected  by  either  immi¬ 
gration  or  emigration. 

The.  errors  involved  in  this  assumption  are  of  small  moment  com¬ 
pared  with  probable  errors  of  observation,  and  vanish  when  the  inter¬ 
vals  of  age  are  taken  exceedingly  minute,  and  where  the  excess  or 
deficiency  of  the  deaths  in  the  former  half  of  the  period  of  time,  with 
reference  to  half  the  deaths  of  the  entire  period,  is  exactly  counter¬ 
balanced  by  a  corresponding  deficiency  or  excess  of  the  deaths  in  the 
latter  half  of  that  period  of  time. 

We  adopt  this  hypothesis,  and  assume  that  each  of  the  ratios  in 
the  column  headed  Mortality  is  identical  with  that  which  would  have 
resulted  had  the  population  of  Prussia  within  the  limits  of  the  ages 
been  stationary  for  a  period  of  years  equal  to  the  specified  interval  ; 
and  we  also  assume  the  accuracy  of  the  Prussian  mortality  and  pop¬ 
ulation  returns. 

From  these  ratios  we  now  proceed  to  determine  duplicate  logarithms 
of  the  probability  that  one  surviving  the  earlier  age  in  each  interval 
will  attain  the  later. 

Let  P0lx  =  the  number  living  under  age  a?,  in  a  stationary  popula¬ 
tion,  in  which  the  same  law  of  mortality  prevails  as  in 
Prussia. 

D0/x  =  the  number  of  annual  deaths  under  age  x  in  the  station¬ 
ary  population. 

Z0  =  the  number  born  alive  each  moment  of  time ,  in  the  stationary 
population. 

lx  —  the  number  surviving  x  years,  out  of  (/0)  the  momentary 
number  of  births. 


60 


A.  MATHEMATICS  AND  PHYSICS. 


j)alb  —  j  —  the  probability  that  one  surviving  the  earlier  age  («) 
will  attain  the  later  (b). 

Kit  —  h  —  4  —  the  number  dying  in  x  years  out  of  (70)  the  mo¬ 
mentary  number  of  births. 

In  a  stationary  population 


and 

therefore, 


n  —  0/x 
Vq,x~  dx' 


d  Po/x  -  ^ x‘l 


=f  (l0  -  K.)  =  ~f\.- 

But  Malb ,  the  ratio  of  the  average  annual  number  of  deaths  in  Prus¬ 
sia  between  ages  a  and  b  to  the  number  of  the  living  between  these 
ages,  computed  with  reference  to  the  middle  of  the  period  in  which 

the  deaths  occur,  equals  ^a/b _ 

•*-alb  -^0 /b  -t-0/a 

Assume  d0/a.  =  Qx  Rx2,  Q  and  R  being  unknown,  and  inde¬ 
pendent  of  the  variable  x. 

Then 

Q  x  -f-  R  x2 


A/,  = 


dx 


and 

Hence 
Ma/b  ^which 


P alb 


_  Pq ib  —  Po/a\  Qb  —  a  -f-  Rb2  —  a 2 

“  Kb-Pj  ~  7  7 _ 

lob  —  a —  *4 — 71 - o 


Q  — j—  R  b  — |—  u 


l0  Q 


b-\- 


R 


b2  -|-  a  b  -|-  a2 


Mr, 


Q  — j—  jR  c  -j-  b 


and 


MATHEMATICS. 


61 


So  also, 

\palb  (which 

and 


^  .  Z0 -  ^0/6 

XT  =  XTT- 


\  _  —  Q&  —  Rb2 

)  ~XlQ—  Qa—Ra^ 


*Pb,c  =  * 


Z0  —  Qc  —  Rc2 
l0—Qb—Rb 2 


Given  Ma/5  and  Mi/C,  required  X^tt/6  and  X  j?6/c. 

First  determine  values  for  Q  and  E  ;  the  values  of  X  palb  and  \pbJC 
are  then  readily  found. 


and 


in  which 


n_y'Ma/b-  p'Mb/c 
^  y'P  —  P'y  *  °’ 


v  y  Ma/b  —  /3  Mble, 

R=  y/3' -  fiy1  -?“; 


0  =  i  + 


ft1  —  b  -[-  a  -(- 


(b*+ba  +  a*)Malb. 

3 


_  (c  +  &)Jf„., 

7  2  -  7 

y  =  c  +  J+(£±5i+^ 

3 

/ 

The  reduction  may  be  simplified  by  letting  Z0  =  1,  and  by  the  use 
of  addition  and  subtraction  logarithms. 

In  like  manner,  from  Mb/C  and  Mc/d  obtain  \pb/c  and  \pc/d ;  and  so 
on  for  all  intervals  of  age  which  the  returns  give. 

We  thus  obtain  duplicate  values  for  the  logarithms  of  the  proba¬ 
bilities  of  surviving  all  the  intervals  specified  except  two.  For  the 
first  and  the  last  interval  we  have  but  single  values.  We  may,  without 
material  error,  adopt  for  the  true  probability  the  mean  of  these  dupli¬ 
cate  results. 

It  will  be  observed  that  the  conversion,  in  each  of  these  cases,  is 
made  for  the  entire  interval,  not,  as  is  more  frequent,  for  the  middle 
6 


62 


A.  MATHEMATICS  AND  PHYSICS. 


year  of  the  interval.  We  are  thus  enabled,  without  the  intervention 
of  a  general  interpolation,  to  compute  directly  the  number  surviving 
at  certain  ages  in  the  resulting  life-table,  out  of  a  specified  number 
born  alive. 

Usually  the  conversion  is  from  a  single  ratio,  based  upon  the 
assumption  of  a  uniform  distribution  of  deaths  throughout  the  interval. 
By  the  present  method,  however,  the  conversion  is  effected,  taking 
into  account  the  actual  or  variable  distribution  of  deaths,  from  three 
consecutive  ratios,  one  preceding  and  another  following  the  interval. 
A  comparison  of  the  relative  accuracy  and  simplicity  of  several 
methods  for  effecting  the  conversion  will  be  given  on  a  following 
page. 

We  now  proceed  to  indicate  methods  for  obtaining  'probabilities  of 
surviving  from  birth  to  ages  one,  three ,  and  five. 

We  have  the  average  annual  number  of  deaths  in  Prussia  under 
the  ages  of  one,  three,  and  five  (jD0/1,  D0/ 3,  D0/ 5),  for  the  period  of  the 
three  years  1839,  ’40,  ’41  ;  and  the  population  under  the  age  of  five 
(P0/5)  at  the  end  of  the  middle  year  of  the  period  (end  of  1840) ; 


also  the  ratio 


© 


of  the  annual  increase  in  the  number  of  births 

deduced  from  the  numbers  registered  for  each  of  the  six  years 
1836-41.  The  average  annual  number  of  deaths  for  the  three 
years  1839,  ’40,  ’41  we  shall  consider  identical  with  the  number  of 
deaths  for  the  year  1840. 

From  the  following,  it  would  appear  that  the  accurate  number  of 
those  born  alive  cannot  be  obtained  directly  from  official  reports,  be¬ 
cause  of  probable  deficiencies  in  registration.  If  the  numbers  of  the 
living  and  of  the  dying  at  the  earlier  ages  have  been  accurately 
observed  and  returned,  if  the  numbers  at  these  ages  have  been  but 
little  affected  by  immigration  and  emigration,  and  if  the  ratio  of 
annual  increase  in  the  number  of  births  can  be  obtained,  a  close 
approximation  to  the  actual  number  of  those  born  alive  may  be 
computed. 

Let  Z0  be  the  number  of  those  momentarily  born  alive  in  Prussia  at 
the  time  for  which  the  census  was  taken  (end  of  1840). 


the  ratio  of  the  annual 


1  /births  1839,  ’40,  ’41\*__  , 

V  —  (births  1836,  ’37,  ’38/ 
increase  in  the  number  of  births  estimated  from  those  registered  for 
each  of  the  six  years  1336-41. 


MATHEMATICS. 


63 


X  -  =  .0066586,  the  logarithm  of  this  ratio. 
v 

Let  So,*  he  the  number  that  died  before  attaining  the  age  of  x  years 
(according  to  the  prevailing  law  of  mortality)  out  of  (/0)  the  number 
born  alive  in  Prussia  during  the  moment  of  time  (end  of  1840)  that 
the  enumeration  of  the  living  is  supposed  to  have  been  made. 

vx  d  80/3.  will  express  the  number  of  those  aged  x  years  that  died  in 
Prussia  during  the  supposed  moment  of  enumeration. 

/vx  d  80/x  S'  vx  =  Dq/x  , 

0  */  0 

the  annual  number  of  deaths  in  Prussia  under  the  age  of  x  years,  for 
the  year  ending  with  the  census,  i.  e.  for  the  year  1840. 


/ *x  /%x 

vx  I  v~x  I  vxd80/x 

J  0  J  o 


represents  the  total  number  that  died  in  Prussia  during  the  x  years 
preceding  the  time  of  the  enumeration  of  the  living,  out  of  the  num¬ 
bers  born  alive  within  that  period.  This  expression  obviously  equals 


jy 


vx 


represents  the  number  born  alive  during  the  x  years  preceding  the 
time  of  the  enumeration. 

The  numbers  born  alive  within  this  period  of  x  years,  Jess  the  num¬ 
bers  dying  within  the  period  out  of  the  numbers  born  alive,  obviously 
represent  the  numbers  of  the  living  at  the  end  of  the  period  under  the 
age  of  x  years  ;  immigration  and  emigration  among  those  under  age 
x  being  considered  null. 


the  numbers  born  alive  during  (1840)  the  year  immediately  preceding 
the  time  of  enumeration. 


64 


A.  MATHEMATICS  AND  PHYSICS. 


Hence 


l0  (=  z»/V)  =  |p„,. + 


O/x 


>/> 

V> 


*/o  fov*y0*v* 


,  rr  ,  »*  —  1 


in  which  V  is  the  Napierian  logarithm  of  v. 
Let  a?  =  5  ;  then  will 


L0  = 

To  simplify,  let 


75  ,  r  v 5  7  Doudx)  V  —  1 

^  Vo  ^^=1  '  U5  -  1 


Do,-  d  x 


v5  V  D0/x  d , 

v  —  1 


v 


Then 


L0  —  \Po/jt'\- D'o,xdx\  v 5 


The  returns  give 
Dm  =  103,509 

D0i3  =  157,628 

D0i5  =  178,539 


average  annual  deaths  under  ages  one,  three, 
and  five. 


P0/5  =  2,249,284 


population  under  age  five  at  the  end  of  the 
year  1840. 

From  these,  and  from  .0066586  X  the  logarithm  of  the  ratio 

of  annual  increase  among  registered  births,  we  find 
B'm=  98,100, 

D'0l 3  =  154,043, 

D'0/S  =  179,912. 

Assume 


Do,x  =  D'o,  o  [=0]  -[-  a?  0  -|-  a? .  a?  —  1  02  -(-  a? .  a?  —  1  .  a?  —  3  03  -f- 


x  .  x — 1  .  a?  —  3  .  x  —  5  R 

=  a? ^  (a:2 — —  4  a?2-)- 3a?)  03-[-(a?4 — 9a?3 -(-23 a?2 — 15a?)  R. 


MATHEMATICS. 


65 


_x'  i  20.-3  (3*-16)*+18 

2  r+— S- ■ 6  + - 6 - 6 

.  [(12a  —  135) o;  + 460]  o:  —  450  ) 
+ - 30 - R\ 


dJK , 

d  x 


=  0  +  (2a—  l)d2  +  (3a2  —  8a  -f  3)  03 


+  (4a3  —  27 a2  +  46a  —  15)  R. 


=  202-|-  (6  a  —  8)  6s  -f  (12a2  —  54a  +  46)  R. 

yd  X) 

8 ,  8 2,  and  83  are  the  divided  differences  of  the  values  V0l0  (=  0), 
D0/ n  D0/3,  and  D0/5 ;  and  R  is  indeterminate. 


D'oio  — 

D'o/3  - 


A  d 


0  000 
98,100 
154,043 
179,912 


98,100  98,100. 
55,943  27,971.5 
25,869  12,934.5 


A  0  8* 

—  70,128.5  — 23,376.17 

—  15,037.0  —  3,759.25 


Ad2  d3 
19,616.92  3,923.38 


We  observe  that  the  divided  differences  of  the  first  order  are  posi¬ 
tive,  and  that  they  diminish  as  the  age  advances. 

Required  for  R  a  value  such  that  the  first  differential  coefficients 
of  the  function  assumed  for  D'0/x  be  positive.  It  would  also  be  de¬ 
sirable,  if  possible,  that  the  second  differential  coefficients,  from  birth 
to  at  least  age  five,  be  negative. 

The  latter  is  not  possible  for  the  entire  period,  with  our  present 
values  for  Don  ?  D'o#  ?  and  Do/5 ,  if  we  assume  but  one  arbitrary  value 
(R).  Our  object,  however,  is  sufficiently  attained  by  taking,  for  R,  a 
value  such  that  for  ages  three  and  five  the  above  conditions  shall  be 
observed. 

That  the  first  differential  coefficients  be  positive  for  ages  three  and 
five,  it  is  requisite  that 

R  <  396.6 

>  —  920.1; 

that  the  second  differential  coefficients  be  negative  for  the  same  ages, 
it  is  requisite  that 


6 


66 


A.  MATHEMATICS  AND  PHYSICS. 


R  >  —  939.8 
<  —  520.6 ; 

from  which  it  appears  that  R  should  be  negative,  and  that  its  value  be 
between  —  920.1 
and  —  520.6 

Let  R  —  —  700.  We  now  have 

*  $  =  98,100 

62  =  —  23,376.17 

63  =  3,923.38 

R  =  —  700. 


/ 


Hence 

m  j  t  i  .  ,  25  .  175  „  .  325  3  125  _ 

D',lx  d  x  (which  =  —  6  +  —  e  - 12  R)  ~  657^95. 


L0  —  |p0,5  P(J/X  d  Z  !  ^5 

in  which 

X  V  ~  * ■  =  1.3142433, 

i>5—  1 

and 

P0I5  =  2,249,284  ; 

therefore, 

v  —  1 
?—  1  ’ 


L0  =  599,418. 

By  the  above  process  the  probable  number  born  alive  during  the 
year  1840  is  found  to  have  been  599,418  instead  of  562,394,  the 
average  of  the  numbers  registered  as  born  alive  during  each  of  the 
three  years  1839,  ’40,  ’41  ;  thereby  indicating  an  annual  deficiency 
in  the  registration  of  37,024,  or  about  6.2  per  cent  of  the  probable 
number  born. 

In  the  above  we  have  supposed  the  numbers  of  the  dying  and  of  the 
living  at  early  ages  accurately  returned.  If  either  be  represented  less 
than  truth,  the  resulting  correction  would  give  still  larger  the  probable 
number  of  births.  Correction  for  deaths  that  escape  registration,  if 
any,  would  tend  to  reduce  the  probabilities  of  living. 

v _ j  i 

Having  found  L0  (which  equals  — y —  .  the  annual  number 

of  births  for  the  year  1840,  we  next  seek  values,  corresponding  to 
intervals  of  age  0-1,  1-3,  and  3-5,  for  D"0/x  (which  equals 
v  —  1  (t , 


d  x 


the  annual  number  of  deaths  in  a  stationary  popula- 


MATHEMATICS. 


67 


tion  in  which  L0  is  the  annual  number  of  births  ;  or  the  number  that 
must  die  in  x  years,  according  to  the  law  of  mortality  prevailing  in 
Prussia,  out  of  £0>  born  alive. 

v5  V 

vx  D'o/x  ——  ~  ^  -Do ix 

v5V  rx  „  d\,x  v  —  1 


—  -  ivx 

v  —  1  J  o  dx  V 

v5V  , 

—  - .  /  Vx  d 

v  —  1  J  o 


T)u 

0 IX  • 


V  -  1 


vx  d  D"o,x  =  ^y-  *  d  ivV  D'o /*)• 


But 


d  ( vx  .  D'0/x)  =  vx  .  d  D'o/x  -|-  D'o/x  •  d  vx  —  vx  (d  D'0/x  -f-  V  D'0,x  d  x) ; 

v  —  1 


..  dD\lx=  {dD'0/x+  VD'0/Xdx) 


v5  V 


Integrating, 


W^+V-/.  D'°’*dx- 

V ,  V,  D' o/i,  jD'o/3?  an(^  -D'o/s  are  already  known;  also  0,  02,  and  03, 
and  I?  in  the  expression 


y*x  ,  X^  (  .  , 


2*-3  (3*-  16)*+  18 

+ - g - * 


.  [(12  x  —  135)  a?  +  460]  x  —  450  _ 
30 


Substituting  for  x  values  1,  3,  and  5,  we  have 
J'1^D!0!Xdx  —  55,899, 

D'o/x  d  x  =  323,335, 


D'0lxdx  =  657,995. 


D,y0/i  =  104,184, 
D"m  =  159,735, 
D\5  =  181,955. 


Therefore, 


68 


A.  MATHEMATICS  AND  PHYSICS. 


X^o/x’  (the  logarithm  of  the  probability  that  one  born  alive  will  sur¬ 
vive  x  years)  —  X - — — ^  =  X  — — -  ^  0/x. 

Iq  ho 

Therefore, 

\pM  =  T.9 170804  =  X  .82619, 

Xp0/3  =  1.8654097  =  X  .73352, 

\p0/5  =  1.8428880  =  X. 69645. 

Hence  of  100,000  born  alive  there  will  attain  the  age  of 
one  year  82,619, 
three  years  73,352, 
five  years  69,645  ; 

or  of  100,389  born  alive  there  will  attain  the  age  of 
one  year  82,941, 
three  years  73,637, 
five  years  69,916. 

The  latter  results  are  those  adopted  in  the  accompanying  inter¬ 
polated  and  other  tables.  These  tables,  as  first  constructed,  repre¬ 
sented  the  probability  of  surviving  five  years  from  birth  to  be  .69916, 
computed  by  a  process  less  rigorous  and  satisfactory  than  the  one 
just  described.  By  assuming  the  same  number  surviving  at  age  five 
(69,916)  as  in  the  original  table,  modification  of  the  values  for  ages 
greater  than  five  becomes  unnecessary. 

The  logarithms  of  the  numbers  surviving  certain  ages  out  of  100,389 
born  alive  may  be  continued  for  ages  greater  than  five,  by  successively 
adding  to  4.8445759  (the  logarithm  of  the  number  surviving  age  five), 
the  logarithms  that  have  previously  been  determined  for  the  proba¬ 
bilities  of  surviving  the  consecutive  intervals. 

The  table  will  then  be  ready,  either  for  a  general  interpolation  of 
the  numbers  surviving  each  anniversary  of  birth,  or  for  obtaining,  by 
abridged  methods,  the  accurate  average  duration  of  life,  life  annuities, 
annual  premiums,  single  premiums,  and  other  practical  tables  involv¬ 
ing  life  contingencies,  for  certain  ages,  without  the  intervention  of  a 
general  interpolation.  Simple  rules  may  also  be  added  for  computing 
from  these  periodical  results  any  specified  values  intermediate. 

The  following  is  a  brief  method  for  finding  approximate  values  for 
the  probabilities  of  surviving  the  intervals  from  birth  to  ages  one, 
three,  and  five,  on  the  supposition  of  a  probable  deficiency  in  the 
registered  number  of  births,  and  that  the  ratio  between  the  numbers 
registered  and  the  true  numbers  is  constant. 


MATHEMATICS. 


69 


The  same  interpretation  of  symbols  is  observed  as  in  the  last 
demonstration. 

We  already  have 

U  =  7.  f  +  =  P°'5+/o , 

Jo  /  «*  0 


■D'o/«  d  * 


f'oV*  ’ 


and 


A>/*  =  flV*  v *• 

When  the  interval  (0-a;)  is  not  large,  xv2  is  a  close  approximation 
to  the  value  of  j  vx  d  x  ;  hence  the  following  approximate  relations. 

L>  =  7i'*=\p"+S!  **»■**]  54r 

D'ou  =  ^  •  % 

vx 

1  ^  ^0/x  d  Do/a? 

ax  vx 

Let  us  first  seek  an  approximate  value  for  L0. 

It  is  obvious  that 

J*q  D' 0/x  d  X  =  f  D'o/x  d  X  +/  D'o/x  d  x  +/  D  o  ix  d  x. 

Assuming  each  term,  in  the  right-hand  member,  to  be  the  integral 
of  the  general  term  of  an  equidijferent  progression,  we  have 

y^D'0(, dx  =  5  —  3  JP'C'5  + 
f*D’IIUdx  =  3^1 
/>„„<**=  1=0^!. 

(*  D'o/x  dx  =  D'o, 5  -j-  2  D'q/3  §  D'on  • 

*/  o 


Therefore, 


Since 


_  v5  D0/x 
V°,x  ~  vl  *  l5"’ 

D1 0/l ,  D1 0/3,  and  D^g  equal  respectively  98101,  154044,  and  179913. 


70 


A.  MATHEMATICS  AND  PHYSICS. 


Therefore, 

But 


L.= 


fS D'aix  dx  =  635,152. 

J  0 

P 0/5  1  .Xo  D'o/xdx 

5v 2 

2,249,284  -f  635,152  =  2,884,436 

5  v2 


=  594,851. 

594,851,  the  computed  number  of  births  for  the  year  1840  by  this 
approximate  method,  is  less  by  about  three  fourths  of  one  per  cent 
than  599,418,  the  corresponding  number  of  births  computed  by  the 
previous  method. 

Having  found  an  approximate  value  for 

*  •  r*  (°r  x»>’ 

we  next  wish  approximate  values  for 


rf  .  ^  (or  rx  ^), 

dx  J  o  vx  J 


corresponding  to  intervals  of  age  0-1,  0-3,  and  0-5. 
When  the  interval  b  -  a  is  small, 

6  d  B0lx  ,  ,  D0lb  -  D 


/ 


nearly  equals 


b  +  a  ? 
,  2 


Da 

l 

V 


0/a  JS a,b  # 

b+a * 


Hence  the  following  approximations  : 


vi  V  _  Aw  _  104  306. 

dx  v * 


dx 


D, 


=  55,804. 


==  =  22,234. 

da;  v4 


=  104,306. 
dx 

vi^=  160,110. 
dx 

vi^  =  182,344. 
dx 


*  This  approximation  was  adopted  by  Dr.  Farr  in  constructing  his  Austrian 
Life-Table.  —  Rep.  Reg.  Gen. 


MATHEMATICS. 


71 


X-1 1.9162709  =  .82463. 


X-1 1.8638226  =  .73084. 

§0/5 

-L'o - ~n  \ 

pm  (which  =  - )  =  X-lT.8410233  =  .69346. 

'■"0 

Then  of  100,000  bom  alive  there  will  be  living  at  ages 
(1)  82,456, 

(3)  73,084, 

(5)  69,346  ; 

or  assuming  the  number  living  at  age  five  to  be  69,916,  the  same  as 
in  accompanying  tables,  then  out  of  100,821  born  alive  there  will  be 
living  at  ages  (1)  83,143, 

(3)  73,684, 

(5)  69,916. 

This  table,  joined  with  the  interpolated  for  ages  greater  than  five, 
gives  for  average  future  duration  of  life  36.51  years  from  birth, 
instead  of  36.66,  according  to  the  values  previously  obtained,  and 
adopted  in  the  interpolated  and  other  tables. 

If  we  substitute  562,394,  the  average  annual  number  returned  as 
born  alive  in  Prussia  for  a  period  of  time  (1839,  ’40,  ’41)  of  which 
the  year  1840  was  the  middle,  for  594,851,  the  approximate  number 
just  computed,  we  shall  find 

\pm  =  1.9109082  =  X  .81453, 

X  p0l3  =  1.8544920  =  X  .71531, 

Xp0l5  =  1.8298000  =  X  .67577  ; 
and  out  of  103,461  bom  alive  84,272  will  survive  one  year, 

74,006  “  three  years, 

69,916  u  five  years, 

and  the  average  future  duration  of  life  from  birth  will  appear  to 
have  been  35.61  years. 

A  general  interpolation  of  the  logarithms  of  the  proportions  surviv¬ 
ing  each  anniversary  of  birth  intermediate  the  specified  ages,  gives 
the  following. 


Hence 


Pm  (which  = 


•  Pm  (which  = 


L0 

dx 

L0  ' 

T 

§0/3 

-^0 

d  x 

L0 


72 


A.  MATHEMATICS  AND  PHYSICS, 


Prussian  Life-Table,  calculated  from  the  ages  of  those  dying  during 

THE  THREE  YEARS  1839,  ’40,  ’41  ;  AND  FROM  THE  AGES  OF  THE  LIVING 
COMPUTED  WITH  REFERENCE  TO  THE  MIDDLE  OF  THE  YEAR  1840. 


Logarithms 

Differences 
between 
Consecutive 
Logarithms  of 
the  Probability 
of  Living. 

.  Persons 

Average 
Future 
Duration 
(or  Expec¬ 
tation)  of 
Life. 

of  the  Numbers 
Born,  and  Living 
at  each  Age. 

of  the  Proba¬ 
bility,  at  each 
Age,  of  Living 
One  Year. 

Born,  and 
Living  at 
each  Age. 

Dying 
during  each 
Year  of 
Age. 

Ages. 

IL 

lLx\\  —  lLx 

kPx—*Px+l 

r 

L  —L 

x  *+1 

# 

IP 

X 

*Px 

-4*Px 

JU 

Dx 

Hj 

X 

0 

5.001688 

.917080  —  1. 

—  51797 

100,389 

17,448 

36.66 

1 

4.918768 

.968877 

—  10576 

82,941 

5,736 

2 

4.887645 

.979453 

—  7597 

77,205 

3,568 

3 

4.867098 

.987050 

—  3378 

73,637 

2,163 

4 

4.854148 

.990428 

—  1854 

71,474 

1,558 

5 

4.844576 

.992282 

—  1678 

69,916 

1,232 

47.06 

6 

4.836858 

.993960 

—  1241 

68,684 

948 

7 

4.830818 

.995201 

—  886 

67,736 

745 

8 

4.826019 

.996087 

—  601 

66,991 

601 

9 

4.822106 

.996688 

—  381 

66,390 

504 

10 

4.818794 

.997069 

—  210 

65,886 

443 

44.81 

11 

4.815863 

.997279 

—  84 

65,443 

409 

12 

4.813142 

.997363 

4 

65,034 

393 

i  13 

4.810505 

.997359 

63 

64,641 

392 

14 

4.807864 

.997296 

99 

64,249 

399 

15 

4.805160 

.997197 

120 

63,850 

411 

41.17 

16 

4.802357 

.997077 

121 

63,439 

425 

17 

4.799434 

.996956 

123 

63,014 

440 

18 

4.796390 

.996833 

114 

62,574 

455 

19 

4.793223 

.996719 

107 

62,119 

468 

20 

4.789942 

.996612 

101 

61,651 

479 

37.54 

21 

4.786554 

.996511 

95 

61,172 

489 

22 

4.783065 

.996416 

95 

60,683 

499 

23 

4.779481 

.996321 

100 

60,184 

508 

24 

4.775802 

.996221 

106 

59,676 

517 

25 

4.772023 

.996115 

118 

59,159 

527 

34.02 

26 

4,768138 

.995997 

121 

58,632 

538 

27 

4.764135 

.995876 

125 

58,094 

549 

28 

4.760011 

.995751 

*  128 

57,545 

560 

29 

4.755762 

.995623 

133' 

56,985 

571 

30 

4.751385 

.995490 

137 

56,414 

583 

30.55 

31 

4.746875 

.995353 

140 

55,831 

594 

32 

4.742228 

.995213 

144 

55,237 

606 

33 

4.737441 

.995069 

151 

54,631 

617 

34 

4.732510 

.994918 

153 

54,014 

628 

35 

4.727428 

.994765 

158 

53,386 

640 

27.14 

36 

4.722193 

.994607 

159 

52,746 

651 

37 

4.716800 

.994448 

161 

52.095 

661 

38 

4.711248 

.994287 

165 

51,434 

673 

39 

4.705535 

.994122 

171 

50,761 

682 

40 

4.699657 

.993951 

185 

50,079 

693 

23.76 

41 

4.693608 

.993766 

204 

49,386 

703 

42 

4.687374 

.993562 

230 

48,683 

717 

43 

4.680936 

.993332 

258 

47,966 

731 

44 

4.674268 

.993074 

291 

47,235 

747 

45 

4.667342 

.992783 

323 

46,488 

766 

20.40 

46 

4.660125 

.992460 

352 

45,722 

787 

47 

4.652585 

.992108 

391 

44,935 

809 

48 

4.644693 

.991717 

438 

44,126 

834 

49 

4.636410 

.991279 

498 

43,292 

860 

50 

4.627689 

.990781  —1. 

570 

42,432 

892 

17.11 

MATHEMATICS, 


73 


Ages. 

Logarithms 

Differences 
between 
Consecutive 
Logarithms  of 
the  Probability 
of  Living. 

Persons 

Average 
Future 
Duration 
(or  Expec¬ 
tation)  of 
Life. 

Of  the  Numbers 
Born,  and  Living 
at  each  Age. 

Of  the  Proba¬ 
bility,  at  each 
Age,  of  Living 
One  Year. 

Born,  and 
Living  at 
each  Age. 

Dying 
during  each 
Year  of 
Age. 

lLx 

lLx+\  —  lLx 

*Ps—*Px+ 1 

L 

X 

^x+\ 

E 

X  I 

~JkPx 

D 

X 

51 

4.618470 

.990211  —  1. 

651 

41,540 

925 

- - —  1 

52 

4.608681 

.989560 

745 

40,615 

965 

53 

4.598241 

.988815 

851 

39,650 

1,008 

54 

4.587056 

.987964 

970 

38,642 

1,057 

55 

4.575020 

.986994 

1101 

37,585 

1,108 

13.98 

56 

4.562014 

.985893 

1307 

36.477 

1,166 

57 

4.547907 

.984586 

1491 

35,311 

1,231 

58 

4.532493 

.983095 

1657 

34,080 

1,302 

59 

4.515588 

.981438 

1803 

32,778 

1,371 

60 

4.497026 

.979635 

1934 

31,407 

1,439 

11.22  ! 

61 

4.476661 

.977701 

2042 

29,968 

1,500 

1 

62 

4.454362 

.975659 

2137 

28,468 

1,551 

63 

4.430021 

.973522 

2215 

26,917 

1,592 

64 

4.403543 

.971307 

2275 

25,325 

1,619 

65 

4.374850 

.969032 

2323 

23,706 

1,632 

9.03 

66 

4.343882 

.966709 

2330 

22,074 

1,629 

67 

4.310591 

.964379 

2337 

20,445 

1,610 

68 

4.274970 

.962042 

2345 

18,835 

1,576 

69 

4.237012 

.959697 

2351 

17,259 

1,530 

70  J 

4.196709 

.957346 

2360 

15,729 

1,471 

7.36 

71 

4.154055 

.954986 

2367 

14,258 

1,404 

72 

4.109041 

.952619 

2414 

12,854 

1,328 

73  j 

4.061660 

.950205 

2828 

11,526 

1,249 

74 

4.011865 

.947377 

2988 

10,277 

1,173 

75 

3.959242 

.944389 

3157 

9,104 

1,094 

5.97 

76  ! 

3.903631 

.941232 

3338 

8,010 

1,014 

77  i 

3.844863 

.937894 

3527 

6,996 

932 

78  j 

3.782757 

.934367 

3726 

6,064 

851 

79 

3.717124 

.930641 

3939 

5,213 

769 

80 

3.647765 

.926702 

4162 

4,444 

690  1 

4.80 

81 

3.574467 

.922540 

4399 

3,754 

613 

82 

3.497007 

.918141 

4648 

3,141 

540 

83 

3.415148 

.913493 

4913 

2,601 

470 

84 

3.328641 

.908580 

5191 

2,131 

404 

85 

3.237221 

.903389 

5485 

1,727 

345 

3.82 

86 

3.140610 

.897904 

5799 

1,382 

289 

87 

3.038514 

.892105 

6126 

1,093 

241 

88 

2.930619 

.885979 

6475 

852 

196 

89 

2.816598 

.879504 

6842 

656 

159 

90 

2.696102 

.872662 

7231 

497 

127 

3.02 

91 

2.568764 

.865431 

7641 

370 

98 

92 

2.434195 

.857790 

8076 

272 

76 

93 

2.291985 

.849714 

8534 

196 

57 

94 

2.141699 

.841180 

9019 

139 

43 

95 

1.982879 

.832161 

9530 

96 

31 

96 

1.815040 

.822631 

10072 

65 

22 

97 

1.637671 

.812559 

10644 

43 

15 

98 

1.450230 

.801915  . 

11248 

28 

10 

99 

1  252145 

.790667 

11887 

18 

7 

100 

1.042812 

.778780 

12562 

11 

4.4 

101 

0.821592 

.766218 

13275 

6.6 

2.7 

102 

0.587810 

.752943 

14029 

3.9 

1.7 

103 

0.340753 

.738914 

14826 

2.2 

1.0 

104 

0.079667 

.724088  —  1. 

1.2 

.6 

105 

1.803755 

1 

.6 

.6 

7 


74 


A.  MATHEMATICS  AND  PHYSICS. 


The  leading  features  in  the  interpolated  Life-Table  for  Prussia 
are  two. 

1st.  Strict  conformity  at  certain  points  to  values  calculated  from 
actual  data. 

2d.  Regularity  in  the  graduation. 

It  will  be  observed  that  the  logarithms  of  the  proportions  born  alive 
and  surviving  ages  1,  3,  5,  14,  25,  35,  45,  55,  65,  75,  and  85,  as 
calculated  from  actual  data,  are  identical  with  those  in  the  interpolated 
table.  More  frequent  coincidence  would  fail,  for  certain  intervals  of 
age,  to  secure  the  desired  regularity. 

From  these  values  we  find,  by  inspection,  that  the  logarithms  of 
the  reciprocals  of  the  probabilities  of  surviving  equal  consecutive  in¬ 
tervals  of  age  diminish  from  birth,  until  they  attain  a  minimum  between 
ages  5  and  25  (near  age  14),  then  gradually  increase  for  subsequent 
intervals. 

In  effecting  the  interpolation,  we  sought  to  arrive  only  at  results 
that,  coinciding  at  the  ages  above  specified  with  those  derived  from 
the  actual  data,  should  represent  the  logarithms  of  the  reciprocals  of 
the  probabilities  of  surviving  consecutive  annual  intervals  of  age  as 
diminishing  from  birth  to  a  minimum  at  some  point  between  ages 
5  and  25,  then  gradually  increasing  for  subsequent  intervals  of  age  ; 
and  that  the  differences  between  these  logarithms  should  also  advance 
without  manifest  irregularity,  increasing  from,  at  latest,  age  25  to 
extreme  old  age. 

Two  distinct  functions  of  interpolation  were  employed. 

1st.  The  exponential. 

For  X  Lx,  write  <px. 


0*  =  <{>a  +  (06  —  0a) 


1 


r~u—  i 

in  which  0a ,  06 ,  0C  are  known  values  of  the  function  0* ,  correspond¬ 
ing  to  ages  a ,  &,  and  c.  q  is  to  be  determined.  If  the  terms  be 
equidistant ,  that  is,  if  c  —  b  —  b  —  a , 


q  — 


and 


0c  ~~  06 
06-0/ 


0*  —  0a  + 


(06  —  0a)2 


(0c  —  06)  —  (06  —  0a) 


.  (qx~a- 1). 


If  the  terms  be  not  equidistant,  the  determining  of  q  will  involve 
the  solution  of  quadratic  or  higher  equations. 


MATHEMATICS. 


75 


2d.  The  algebraic. 

<t>x- 

in  which 


*.T  + 


Br  ,  ,  C,  ,  D, 


+  Qn, ; 


rL.  =  #  —  a  .  x  —  b  .  x  —  c  .  x  —  d 


and 


n. 


,  Bx  = 


n*  r  _  n, 

—  b'Lx  X  — 


Dx  =  - 

C  X  - 


u. 


d  ’ 


Q  may  be  zero,  or  an  arbitrary  constant  real  and  finite,  or  a  real 
function  involving  only  integral  powers  of  the  variable,  and  which 
cannot  cause  the  term  (Qnx)  to  become  infinite  or  indeterminate  for 
any  value  of  the  variable  within  the  limits  assigned  for  interpolation, 
or  between  those  corresponding  to  the  extreme  given  values  of  the 
function. 

Ax  (  obviously  becomes  unity ,  and  terms  indepen- 


When  x  = 


’  Aa  (  dent  of  this  factor  vanish. 


h 


T> 

_L  a  u  t& 

B„ 

n 

r  x  CC  U  « 

’  c: 

rJ  u  u  u 

’  Da 

Another  convenient  function  for  interpolation  when  three  terms  are 
given,  but  which  was  not  employed  in  framing  the  present  table,  is 
the  general  parabolic. 


in  which 


—  tya  -(“  ($6  -  $«)  ^ - > 

.  <t>c~  Qa 


 <f>b  — 
c  —  a 

A  - - 


The  exponential  involves  but  three  known  values  of  the  function. 
The  number  of  known  values  that  may  enter  into  the  interpolation 
by  the  algebraic  is  unlimited ;  but,  without  care,  the  resulting  series 
will  often  be  quite  eccentric. 

Given,  logarithms  of  the  proportions  born  alive,  and  surviving  ages 


76 


A.  MATHEMATICS  AND  PHYSICS. 


1,  3,  5,  14,  25,  35,  45,  55,  65,  75,  and  85 ;  required  (X  Lx)  the 
logarithms  of  the  proportions  surviving  each  intermediate  anniversary 
of  birth. 

By  the  exponential  formula  values  between  ages 

25  and  38 1  .  .  ,  ,  „  f  25,  35.  and  45. 

35  d  48  !  were  resPectlvely  interpolated  from  j  ’  ’ 


.  known  values  of  the  function  for  ^  35,  45,  and  55. 

45  and  58  |  ao.eg  |  45,  55,  and  65. 

55  and  68  J  [  55,  65,  and  75. 

Values  from  73  to  age  105,  inclusive,  were  interpolated  from  known 
values  of  the  function  for  ages  65,  75,  and  85. 

By  the  above  it  appears  that  duplicate  values  were  obtained  at  ages 
36  and  37,  46  and  47,  56  and  57. 

The  results  deduced  from  these  interpolations  conform  strictly  to  the 
conditions  imposed,  except  near  the  joining  points  of  the  several  series, 
where  appear  irregularities  in  the  first  and  second  orders  of  differences. 
These  manifest  irregularities  were  then  corrected  for  the  several  in¬ 
tervals,  by  adding  to  the  result  at  each  of  the  several  ages  the  corre¬ 
sponding  value  of  A,,  derived  from  the  simple  algebraic  function 


A,  = 


x  —  a  .  x  —  b  .  x  —  c  .  x  —  h 


g  a  .  g  b  .  g  —  c.g 


h 


+ 


a  .  x  —  b  .  x  —  c  .  x  —  g 
a  .  h  —  b  .  h  —  c  .  h  —  g 


In  applying  the  correction  to  ages  between  35  and  48,  a,  Z>,  c,  g , 
and  h  equalled  respectively  35,  45,  55,  36,  and  37.  A:6  and  A37  were 

the  differences  between  the  duplicate  values  for  ages  36  and  37. 
The  difference  was  positive  when  the  one  of  the  duplicate  values 
first  obtained  was  the  greater. 

In  correcting  between  ages  45  and  58,  a ,  Z>,  c,  g ,  and  li  equal  re¬ 
spectively  45,  55,  65,  46,  and  47,  and  A46  and  A17  were  the  differences 
respectively  between  the  X  L46  and  X  L„  just  corrected,  and  the  corre¬ 
sponding  results  derived  by  the  exponential  formula  from  values  for 
ages  45,  55,  and  65. 

By  a  similar  process  the  correction  was  made  for  values  between 
ages  55  and  68. 

A  slight  irregularity  still  existing  in  the  second  differences  (the 
first  differences  from  the  logarithms  of  the  probability  of  living)  near 


MATHEMATICS. 


77 


the  joining  of  the  series  about  age  36,  another  correction  was  made  to 
the  values  between  ages  37  and  45,  viz.  : 

x  —  35  .  x  —  3 6.x  —  37  .  x  —  45  .  x  —  46  .  x  —  47 

=  34  — 35 . 34  —  36 . 34  —  37 . 34  — 45 . 34—46 . 34  —  47' 

The  values  for  A34 ,  being  the  difference  between  the  first  of  the 
duplicate  \L3i  and  the  corrected  second  of  the  duplicates.  Ax  is 
additive,  if  the  first  of  the  duplicate  values  for  X  L3i  is  the  greater. 

The  values  between  ages  67  and  73,  inclusive,  were  computed  from 
the  known  values  at  ages  65,  66,  67,  and  73,  by  assuming  the  third 
order  of  differences  constant. 

X  L73  =  X  L65  -]—  8  A  — |-  28  A2  -j—  56  A3 . 

A  and  A2  were  derived  from  the  original  value  for  X  L65 ,  and  from 
the  corrected  values  for  X  L66  and  XL67.  a3  was  then  readily  found, 
and  consequently  the  values  required  between  ages  67  and  73.  A 
modification  of  the  method  here  indicated  might  have  been  applied 
with  advantage  to  the  correction  of  irregularities  near  the  points  of 
junction  in  other  parts  of  the  table. 

From  the  given  values  of  X  Lx  for  ages  3,  5,  14,  25,  35,  together 
with  the  values  of  X  Lx  for  ages  26  and  27,  computed  as  above,  the 
unknown  values  between  ages  5  and  26  were  interpolated  by  the 
algebraic  formula 

X  Lx  =  0,  =  $3  +  $5  -gj  +  $14  ~f"  $25  +  $26  jjr  + 

,  F*  ,  ,  G. 

$27  -ft-  T-  $35  ~FT' 

-^27  ^35 

The  forms  of  the  functions  A ,  B,  C,  &c.  have  been  previously  given. 

From  XLmXL3,  and  X  L5  values  were  deduced  by  the  exponential 
formula  for  X  L2  (=  4.887645)  and  X  ] b4  (=  4.854456). 

By  the  same  formula,  from  X  L3,  X  L5,  and  the  computed  value  for 
\L7  was  deduced  a  duplicate  value  for  x£4  (=  4.853532).  From 
comparison  of  the  duplicate  values  for  \L 4,  giving  to  the  former 
double  weight,  we  obtain  4.854148. 

We  remark  that  the  desired  regularity  in  the  graduation,  for  the 
greater  part  of  the  table,  was  attained  by  making  identical  three  or 
more  consecutive  values  of  adjoining  series. 

It  will  be  observed  that  the  interpolated  results  represent  mortality 
1* 


78 


A.  MATHEMATICS  AND  PHYSICS. 


diminishing  from  birth,  until  attaining  a  minimum  about  age  12,  then 
increasing  gradually  to  age  105,  the  assumed  terminating  age  of  the 
table.  Also,  that  the  values  in  the  column  of  differences  headed 
—  A  \px  gradually  increase  through  the  greater  part  of  the  entire 
table,  diminishing,  however,  between  ages  17  and  22.  A  curve,  to 
which  the  intervals  of  age  and  corresponding  intensities  of  mortality 
are  co-ordinates,  will  be  concave  downwards  through  the  space  where 
these  differences  diminish,  if  elsewhere  concave  upwards.  The  attain¬ 
ment  of  regularity  at  joining  points  in  the  order  of  differences  next 
higher,  was  deemed  unimportant.  For  the  accuracy  with  which  much 
of  the  arithmetical  computation  has  been  performed,  in  the  preparation 
of  this  and  certain  other  tables  following,  credit  is  due  to  Mr.  Howard 
D.  Marshall,  of  Boston.* 

Life-Tables,  advancing,  by  regular  gradations,  from  birth  to  extreme 
old  age,  and  conforming  strictly  at  convenient  intervals  to  values  de¬ 
rived  from  original  data,  are  uncommon. 

The  graduation  of  the  older  tables  was  very  imperfect.  The  Car¬ 
lisle  gives  the  annual  rate  of  mortality  at  age  20  greater  than  at  23 ; 
at  31,  greater  than  at  34 ;  at  46  greater  than  at  51 ;  at  88  greater  than 
at  89  ;  and  at  91  the  same  as  at  101.  Mr.  Milne’s  excellent  table  for 
Sweden  and  Finland,  (1801-5,)  though  less  faulty,  is  still  irregular; 
so  also  those  of  De  Parcieux,  Kersseboom,  Finlaison,  and  others. 

The  valuable  and  elaborate  English  Life-Tables  prepared  by  Dr. 
Farr,  and  published  in  the  Reports  of  the  Registrar-General  (Eng¬ 
land),  and  also  the  one  prepared  by  a  committee  of  eminent  actuaries 
to  represent  a  law  of  mortality  according  to  the  combined  experience 
of  Insurance  Companies,  as  published  by  Mr.  Jenkin  Jones,  vary  the 
results  derived  from  actual  data,  to  conform  to  assumed  laws.  The 
graduation  of  the  Actuaries’  Table  is  unexceptionable  ;  that  of  the 
tables  of  Dr.  Farr  nearly  so. 

The  important  tables  presented  by  Mr.  E.  J.  Farren,  in  his  instruc¬ 
tive  treatise  entitled,  “  Life  Contingency  Tables,  Part  I.,”  begin 
with  age  21,  and  conform  strictly  at  decennial  points  to  values  derived 
from  actual  data.  The  function  of  interpolation  adopted  by  him  was 
the  Calculus  of  Finite  Differences,  so  far  as  possible  ;  assuming,  how¬ 
ever,  the  intensity  of  mortality  to  advance  by  a  constant  ratio,  when, 
either  from  paucity  of  data  or  other  sufficient  cause,  the  Calculus  of 


*  Mr.  Marshall  has  deceased  since  this  paper  was  prepared,  and  in  press. 


MATHEMATICS. 


79 


Finite  Differences  was  inapplicable.  The  results  attained  are  entire¬ 
ly  regular. 

Many  writers  on  this  subject  have  felt  it  desirable  that  some 
simple  generic  law  be  discovered,  which,  by  suitable  changes  in  the 
constants,  will  approximate  the  specific  laws  of  human  mortality  indi¬ 
cated  by  known  tables.  Among  the  more  philosophical  conceptions 
is  the  one  of  Mr.  Gompertz  (Philosophical  Transactions,  1825),  that 
for  the  greater  part  of  life  man  momentarily  loses  44  equal  proportions 
of  his  remaining  power  to  oppose  destruction  ” ;  and  consequently, 
that  the  intensity  of  mortality  increases  with  advancing  age  by  a  con¬ 
stant  ratio.  Mr.  Edmonds  would  have  u  the  force  of  mortality  at  all 
ages  ”  44  expressible  by  the  terms  of  three  geometric  series,  so  con¬ 
nected  that  the  last  term  of  one  series  is  the  first  of  the  succeeding 
series.”  Dr.  Farr  recognized  the  principle  in  framing  his  English 
Tables  for  1841;  treating  44  the  two  series  of  numbers  representing 
the  mortality  from  15  to  55,  and  from  55  to  95,  as  geometrical  pro¬ 
gressions.  The  ratios  were  derived  from  a  comparison  of  the  increase 
in  the  mortality  at  15-20,  25-30,  35-40,  &c. ;  and  the  increase  at 
20-25,  30-35,  40-45,  &c. ;  and  the  first  terms  were  derived  from 
these  ratios,  and  the  sums  of  the  series  which  they  formed.” 

Mr.  Orchard’s  method,  as  described  by  Mr.  Gray  in  the  Assurance 
Magazine,  (London,)  for  July,  1856,  was  the  adoption  of  44  two  con¬ 
secutive  series,  having  constant  second  differences,”  to  represent  the 
proportions  living  from  age  20  to  80  and  from  80  to  96,  the  ter¬ 
minating  age  of  his  table.  He  wished  44  to  find  a  simple  algebraical 
relation  which  should  passably  well  represent  some  of  our  best  tables.” 
The  advantage  claimed  for  a  table  so  constituted  44  is,  that  it  admits, 
by  the  application  of  simple  analytical  processes,  of  the  independent 
formation  of  any  of  the  values  which  ordinarily  require  the  aid  of  a 
formidable  array  of  the  results  of  previous  computation.”  The  same 
paper  gives  a  single  algebraical  function  of  the  second  degree  pro¬ 
posed  by  Mr.  Babbage,  which  is  said  to  represent,  nearly,  the  Swedish 
Table  of  Mortality. 

Other  methods  have  been  proposed  by  mathematicians  of  estab¬ 
lished  reputation.* 

*  A  valuable  contribution  to  this  department  of  the  science  of  Vital  Statistics 
was  read  before  the  American  Association,  at  its  late  meeting,  by  President  McCay 
of  South  Carolina. 


80 


A.  MATHEMATICS  AND  PHYSICS. 


15.  Discussion  of  Certain  Methods  for  converting  Katio  of  Deaths 
to  Population,  within  given  Intervals  of  Age,  into  Logarithms  of 
the  Probability  that  one  living  at  the  Earlier  Age  will  attain 
the  Later.  With  Illustrations  from  English  and  Prussian  data. 


In  the  paper  immediately  preceding,  a  method  has  been  indicated 
for  the  conversion  of  mortality  into  probability  of  living  from  a  com¬ 
parison  of  three  consecutive  ratios,  one  preceding  and  another  follow¬ 
ing  the  specified  interval.  In  the  present  paper  the  results  so  derived 
will  be  compared  with  others  obtained  from  a  single  ratio. 

Let  m  (identical  with  M  in  the  preceding  paper)  represent  the  rate 

of  annual  mortality  for  any  interval  of  age,  or  the  ratio  (t)  of  the 


number  annually  dying  to  the  number  living  in  the  community  within 
that  interval  of  age. 

If  the  population  be  stationary ,  ma/b  (which  equals  f  the  rate  of 

-t-alb 

annual  mortality  for  the  interval  between  ages  a  and  b,  will  equal 


Lxdx  fbaL;dx' 


If  also  the  deaths  be  supposed  uniformly  distributed  throughout  the 

interval  of  age,  i.  e.  — dLx ,  constant,  the  numerator  (/:--•) 

will  represent  the  sum  of  a  series  of  constants,  and  the  denominator 

Lx  dx ^  the  sum  of  a  series  of  values  progressing  by  a  common 

difference  ;  hence  the  value  of  the  fraction  will  be  independent  of  the 
extent  of  the  interval ,  and  will  vary  only  with  the  mean  age. 

If  for  — ,  the  mean  age,  we  substitute  2,  and  assume  for  k 

any  arbitrary  value,  mjzrk,7+z  will  be  constant  for  all  values  of  the 
arbitrary ,  provided  2  k  does  not  exceed  ( b  —  a)  the  limits  of  age 
within  which  the  uniformity  of  distribution  was  assumed. 

It  will  follow  that  y— %  the  value  of  the  probability  that  one  living 

■^z  +  k 


at  the  earlier  age,  2  —  k ,  will  attain  the  later,  2  -f-  k,  expressed  in 
terms  of  the  known  annual  rate  of  mortality  (ffia/i),  and  of  the  arbi¬ 
trary  ( k ),  is 

1  —  k  m0/b 
l+k  malb 


MATHEMATICS. 


81 


Hence  the  probability  of  surviving  the  entire  period  ( b  — a)  is 
_  b  —  a 

1 - 2 —  ma'b 

,  ,  b-a  5 

H - 2~  ma,b 

and  the  probability  of  surviving  the  middle  year  of  the  period  is 

1  —  i  ™alb 

1  +  2-  ma/b 

Again,  if  the  population  be  stationary ,  mxlx~+  d x  .  dx  (for  which  put 
mdx  •  dx),  the  intensity  of  mortality  at  age  x,  or  the  rate  of  momen¬ 
tary  mortality  at  that  age,  will  equal 


—  dLx 

Lz 


=  —  d\  =  —  X 


lx  +  dx 


-  ^  Px/x  +  d  x  ? 


(for  which  put  —  X  p  d3),  the  Napierian  logarithm,  with  the  algebraic 

sign  changed,  of  the  probability  of  surviving  a  moment  of  time  from 
age  x. 

Hence  j*  mdx  d  x,  the  integral  within  the  limits  of  the  ages  a  and 
b  of  the  intensity  of  mortality,  will  equal  — X  pall the  Napierian  log¬ 
arithm,  with  its  sign  changed,  of  the  probability  that  one  living  at  the 
earlier  age  (a)  will  attain  the  later  (b). 

“  A  rate  of  mortality  ”  “  derived  from  the  integration  —  d  L  ” 

has  been  happily  styled  the  “  integral  rate  of  mortality.”  * 

Assuming  deaths  uniformly  distributed ,  md  z  becomes  equal  to  ma/6; 
that  is,  the  rate  of  annual  mortality  at  the  mean  age  equals  the  rate  of 
annual  mortality  for  the  entire  interval ;  consequently 


—  l)Pdz  =  maib  •  d  %  ; 

that  is,  the  Napierian  logarithm,  with  its  sign  changed,  of  the  proba¬ 
bility  of  surviving  a  moment  of  time  at  the  middle  of  the  specified 
interval,  equals  the  rate  of  annual  mortality  for  the  interval,  multiplied 
by  the  differential  of  the  mean  age. 


*  Life  Contingency  Tables,  Part  I.,  by  E.  J.  Earren.  In  the  same  connection  is 
stated  the  important  proposition,  that  “  whatever  progression  prevails  among  the 
integral  rates  of  mortality  at  different  ages,  the  same  progression  will  be  found 
to  prevail  among  the  logarithms  of  the  probabilities  of  living,  and  vice  versa." 


82 


A.  MATHEMATICS  AND  PHYSICS. 


The  intensity  of  mortality  at  age  z  (when  deaths  are  uniformly  dis¬ 


tributed)  being  the  middle  term 


(~dLz 

V  Lz 


^  of  a  series  of  reciprocals 


of  an  arithmetical  progression,  is  less  by  a  small  proportion  than  the 
average  value  of  the  terms  constituting  the  series  ;  hence  ( ma/b )  the 
rate  of  annual  mortality  for  the  interval  of  age  b  —  a  is  somewhat 


less  (in  the  case  of  such  uniform 


distribution)  than 


(/' 


the  integral  rate  of  mortality  for  the  interval,  or  than  its  equivalent 
( — X  'Path ),  the  Napierian  logarithm,  with  the  sign  changed,  of  the 


probability  that  one  living  at  the  earlier  age  (a)  will  attain  the  later 
age  (b). 

To  convert  the  Napierian  to  the  common  logarithm,  we  multiply  by 
p  (=  .4342945),  the  modulus  of  the  common  system. 


PRUSSIA.  1839,  ’40,  ’41. 

Table  comparing  Logarithms  of  Probabilities  of  Surviving,  comput¬ 
ed  by  different  Methods. 


Ratio  of 
Deaths  to 
Population. 

Common  Logarithm,  with  changed  Sign,  of  the  Probability  that  one  liv¬ 
ing  at  the  Earlier  Age  in  each  Interval  will  attain  the  Later. 

Mortality. 

Integral. 

Approximate. 

Ages 

m 

—  Xp 
each 

from  three 
consecutive 
Ratios. 

b  —  a 

1  — m  —— 

x  .  2 

b  —  a 

\  +  m  — 

(b  a)  X  m 

1  2 

—  ( b  —  a)  (u  .  m 

a,  b. 

A 

B 

c 

D 

E 

0-  5 

.0802238 

.157112* 

.176598 

.174297 

.174204 

5-  7 

.0152056 

.013155 

.013208 

.013208 

.013208 

7-14 

.0077790 

.023557 

.023655 

.023649 

.023649 

14-20 

.0062978 

.016416 

.016413 

.016411 

.016411 

20-25 

.0089397 

.019425 

.019416 

.019412 

.019412 

25  -  30 

.0096939 

.021058 

.021054 

.021050 

.021050 

30-35 

.0108317 

.023537 

.023527 

.023521 

.023521 

35-40 

.0131780 

.028637 

.028626 

.028616 

.028616 

40-45 

.0144675 

.031449 

.031430 

.031416 

.031416 

45-55 

.0210345 

.092322 

.091691 

.091355 

.091352 

55-60 

.0357042 

.077981 

.077738 

.077539 

.077531 

60  -  65 

.0557995 

.122189 

.121962 

.121199 

.121167 

65  -  75 

.0909134 

.415608 

.425992 

.395105 

.394832 

75-85 
85  and  i 
upw’ds  \ 

.1515098 

.2661784 

.722021 

.860283 

.659260 

1.162896 

.657999 

1.155998 

*  This  value  was  calculated  by  a  process  described  in  the  preceding  paper,  from 
population  under  age  5  ;  from  deaths  for  the  intervals  of  age  0  -  1 ,  1-3,  and  3  —  5  ; 
and  from  the  rate  of  annual  increase  of  births  estimated  from  registered  births  for 
the  six  years  1836-41. 


MATHEMATICS, 


83 


ENGLAND  AND  WALES. 


Table  comparing  Logarithims  op  Probabilities  of  Surviving,  com¬ 
puted  BY  DIFFERENT  METHODS. 

Deaths  {Seven  Years )  1838-44. 

Population  computed  to  Middle  o/’184l. 

Ninth  Rep.  Reg.-Gen.,  pp.  176,  177. 


Ratio  of 
.Deaths  to 
Population. 

Common  Logarithm,  with  changed  Sign,  of  the  Probability  that  one  living 
at  the  Earlier  Age  in  each  Interval  will  attain  the  Later. 

Mortality. 

Integral. 

Approximate. 

- 

Xp 

m 

Duplicate 

Values, 

Mean  of 

,  b — a 

l-m^T 

i-T 

—  {b—a)j.i .  m 

Ages 
a,  b. 

each  from 
two  con¬ 
secutive 
Ratios. 

the 

Duplicate 

Values. 

.  b~a 
'  +  “-2- 

-{b-a)X^m 

2 

A 

B 

c 

D 

E 

0-  1 

.1792379 

\ 

.077265 

.073073* 

.078052 

.078052 

.077842 

1  -  2 

.0654971 

.028133 

.028379 

.028256 

.028455 

.028455 

.028445 

2-  3 

.0351076 

.015206 

.015235 

.015220 

.015249 

.015249 

.015247 

3-  4 

.0250056 

.010850 

.010854 

.010852 

.010860 

.010860 

.010860 

4-  5 

.0184203 

.007995 

.007998 

.007997 

.008000 

.008000 

.008000 

5-10 

.0091272 

.019686 

.019789 

.019738 

.019823 

.019820 

.019819 

10-15 

.0052572 

.011397 

.011425 

.011411 

.011417 

.011416 

.011416 

15-25 

.0081967 

.035723 

.035643 

.035683 

.035618 

.035598 

.035598 

25-35 

.0098929 

.043033 

.043045 

.043039 

.042999 

.042966 

.042964 

35-45 

.0124582 

.054239 

.054261 

.054250 

.054176 

.054105 

.054105 

45-55 

.0165886 

.072341 

.072636 

.072489 

.072209 

.072044 

.072043 

55-65 

.0295429 

.130216 
j  .130452 

.130334 

.129249 

.128313 

.128303 

65-75 

.0622301 

!  .283777 
j  .278012 

.280895 

.279528 

.270349 

.270262 

75-85 

.1374474 

.718169 
!  .649789 

.683979 

.731961 

.597869 

.596926 

85-95 

.2842092 

1.127822 

1.242716 

1.234305 

9o  — (- 

.4146003 

1 

1.827064 

1.800586 

*  This  value  was  derived  from  the  registered  births  for  the  eleven  years  1839  -  49. 
and  from  the  registered  deaths  under  one  year  of  age  for  the  ten  years  1840-49. 


84 


A.  MATHEMATICS  AND  PHYSICS. 


In  each  of  the  preceding  Tables,  column  A  gives  rates  of  annual 
mortality  for  the  several  specified  intervals  of  age,  or  ratios  of  the 
average  numbers  annually  dying  in  the  community  within  the  speci¬ 
fied  intervals  to  the  numbers  living  within  the  same  intervals,  estimated 
with  y^ference  to  the  middle  of  the  year  or  period  in  which  the  deaths 
occurred. 

Column  B  with  changed  signs  gives  the  common  logarithms  of  the 
probabilities  of  surviving  the  specified  intervals,  each  computed  from 
three  consecutive  ratios  in  the  column  of  mortality  by  a  process  de¬ 
scribed  in  the  preceding  paper.  These  values,  which  we  designate 
integral  values,  may  be  assumed  without  appreciable  error  to  represent 
truly  the  results  demanded  by  actual  data,  and  with  them  may  be  com¬ 
pared  approximate  values  obtained  by  simpler  processes. 

The  approximate  values  in  the  columns  C,  D,  and  E  were  each 
derived  from  single  ratios  in  A. 

The  values  in  C  were  each  obtained  by  first  multiplying  ( m )  the 

annual  rate  of  mortality  by  (t)  half  the  number  of  years  in  the 

interval  of  age ;  then  finding  the  logarithm,  with  changed  sign,  of 
the  quotient  of  unity  less  this  product  divided  by  unity  plus  this 
product. 

The  values  in  D  were  each  found  by  multiplying  the  number  of 
years  in  the  respective  interval  by  the  logarithm,  with  changed  sign, 
of  the  quotient  of  unity  less  half  the  rate  of  mortality  divided  by  unity 
plus  half  the  rate. 

The  values  in  E  were  each  found  by  multiplying  the  mortality  by 
.4342945  (/*),  the  modulus  of  the  common  system  of  logarithms,  and 
by  (b  —  a)  the  number  of  years  in  the  respective  interval  of  age. 

Whenever  the  decrements  in  the  Life-Table  resulting  from  the  origi¬ 
nal  data  are  constant,  the  corresponding  result  in  C  represents  the 
logarithm,  with  changed  sign,  of  the  probability  of  surviving  the  entire 
interval ;  that  in  D  represents  the  product  of  (b  —  a)  the  number  of 
years  in  the  interval,  multiplied  by  the  logarithm,  with  changed  sign,  of 
the  probability  of  surviving  the  middle  year  of  the  interval ;  and  that 

in  E  the  product  of  fc“)  the  number  of  equal  moments  in  the  in¬ 
terval,  multiplied  by  the  logarithm,  with  changed  sign,  of  the  proba¬ 
bility  of  surviving  a  moment  of  time  at  the  middle  of  the  interval. 
Whenever  the  decrements  in  the  Life-Table  are  increasing ,  the  above 


MATHEMATICS. 


85 


results  are  each  less  than  the  respective  logarithm  ;  and  when  de¬ 
creasing ,  greater. 

The  results  in  E  should  in  all  cases  be  somewhat  less  than  those  in 
D,  although  generally  the  approximation  is  so  close  that  values  in  E 
may  without  appreciable  error  be  substituted  for  those  in  ^.J)  * 

The  results  in  D  are  likewise  less  than  corresponding  ones  in  C. 

The  results  in  C  are  less  than  the  integral  values  in  B ,  whenever 
the  decrements  between  the  proportions  surviving  at  equidistant  ages 
in  the  Life-Table,  derived  from  actual  data,  form  a  series  increasing 
with  the  age  ;  they  are  equal  to  them,  when  the  series  is  uniform,  and 
greater  than  truth  when  the  series  diminishes.  By  reference  to  the 
Prussian  Table  interpolated  for  annual  intervals,  we  observe  that  the 
decrements  diminish  from  birth  to  age  13 ;  increase  thence  to  age  65  ; 
and  again  diminish  to  the  age  terminating  the  Table. 

The  process  for  deducing  values  in  D  is  identical  with  that  adopted 
by  Dr.  Farr,*  in  briefly  calculating  approximate  Life-Tables.  After 
determining  from  values  so  obtained  the  proportions  of  the  living  at 
certain  ages,  he  assumed  that  the  proportions  within  the  several  inter¬ 
vals  were  series  in  arithmetical  progression. f  It  is  not  unusual,  in 
framing  Life-Tables  from  population  and  mortality  statistics,  to  let 

i -j 

JL 
2 

interval,  then,  assuming  some  law  of  relation,  to  determine  values 
for  intermediate  ages.  Results  so  deduced  will  commonly  represent 
the  probability  of  living  for  a  large  part  of  life  somewhat  greater 
than  truth  demands. 


aal  the  probability 


C.  Process  for  deducing  accurate  Average  Duration  of  Life. 
Present  Value  of  Life-Annuities,  and  other  useful  Tables  in¬ 
volving  Life-Contingencies,  from  Returns  of  Population  and 
Deaths,  without  the  Intervention  of  a  General  Interpolation. 

The  logarithms  of  the  proportions  surviving  at  certain  ages  (X  Lx) 
are  obtained,  by  successively  adding  to  the  logarithm  of  a  number  as- 


*  To  this  distinguished  writer  the  science  of  vital  statistics  is  largely  indebted 
for  valuable,  extensive,  and  varied  contributions, 
t  Fifth  Report  Reg.- Gen.  Eng.,  p.  362. 

8 


86 


A.  MATHEMATICS  AND  PHYSICS. 


sumed  living  at  birth,  or  other  specified  age,  the  logarithms  of  the 
probabilities  of  surviving  subsequent  intervals.  Processes  for  accu¬ 
rately  and  for  approximately  determining  the  logarithms  of  the  prob¬ 
abilities  of  surviving  have  been  indicated  in  the  previous  papers. 

Average  future  duration  (or  expectation)  of  life  (Ex)  expressed  in 
years  for  any  age  (a?)  may  be  obtained  by  multiplying  by  the  differ¬ 
ential  of  the  age  ( d  x)  the  integral  of  the  proportions  surviving  within 


(/;34 


the  limits  of  the  given  age  and  of  the  greatest  tabular  age 
and  dividing  the  product  by  the  proportions  living  at  the  given  age. 
That  is, 


E„  = 


ixj? L, 


in  which  105  is  assumed  the  greatest  tabular  age. 

A  close  approximation  to  this  value  may  be  found  by  dividing 
lx  =  lx-\~lx+  1  +  . .  .  .  Ll05)  the  sum  of  the  proportions  liv¬ 
ing  at  the  given  age  and  at  each  subsequent  anniversary  by  ( Lx )  the 
proportions  living  at  the  given  age,  and  from  the  quotient  deducting 
the  half  of  unity  ;  that  is, 


Ex  = 


■'£^106  j 

Zj.r 


—  i ,  nearly. 


The  latter  is  the  more  common  process. 

The  formula  expressing  the  value  of  a  life-annuity,  or  the  present 
value  of  one  dollar  payable  at  the  end  of  each  year  during  the  re¬ 
mainder  of  the  life  of  the  annuitant  after  attaining  a  given  age,  is 


Ex+i  v  -j-  Lx+ 2  r2  -j-  .  . . .  -I<io5  x 

~L,  ; 

in  which  v  is  the  present  value  of  one  dollar  due  one  year  hence  at 
a  given  rate  of  interest. 

This  expression  may  readily  be  converted  into  the  well-known 
symmetrical  form 


Lx+l  vx+l  +  Lx+9  vx+z  -!-••••  Lm  v105 
Lxvx 

which  equals 

Lxv*  +  Lx+ 1^  +  1+  Lx+2v*+*+  .  . . .  L!05*105 
Lxvx 


MATHEMATICS. 


87 


Lx  vx 


Given  (L0,  X1?  jL3,  &c.)  the  proportions  born,  and  surviving  ages 
1,  3,  5,  14,  25,  35,  45,  55,  65,  75,  and  85,  according  to  the  law  of 
mortality  prevailing  in  Prussia  ;  required  corresponding  average  future 
duration  of  life,  life-annuities,  and  premiums  annual  and  single. 

/105  ^  106 

Lx  and  Xix  Lxvx,  some  law 
of  relation  must  be  supposed  to  exist  between  the  known  values  of 
each  of  the  functions  Lx  and  Lx  vx  ;  and  this  law  obviously  should 
represent  numbers  diminishing  with  advancing  age. 

The  law  may  either  be  expressed  by  a  single  formula  (as,  for  in¬ 
stance,  the  algebraic  of  the  eleventh  order),  or  by  a  series  of  distinct 
formulae.  In  consequence  of  the  very  great  arithmetical  labor  in¬ 
volved  in  its  practical  application,  it  will  not  often  be  thought  desirable 
to  adopt  a  single  formula. 

When  the  known  values  are  equidistant ,  n  being  the  number  of 
years  in  each  interval,  let 


Sx  —  Lx  +  Lx+n  +  Lx+2n  + - - 


and 

ST. 

—  Lxvx  -}-  Lx+n 

vx+n  4-  ll 

then  will 

/»105  + 

/, 

=/;“ 

and 

'ETL.f 

=  i::+n‘ 

Formulae  which  express  laws  of  relation  supposed  to  exist  between 
four  given  values  we  style  four -point  formulce ;  and  so  for  any  other 
number  of  given  values. 

The  solution  of  the  four -point  algebraic  equation 
X  =  a-\-^X-\-yX2-\~8x3, 


in  which  a,  /3,  y,  8  are  unknown,  and  independent  of  the  variable  (a?), 
may  assume  several  forms  ;  one  of  the  more  convenient  of  which 
for  our  present  purpose  is 


X  =  B  - - -  +  C  —  +  *  —  b  .  x  —  c 

b  —  c  c  —  b 


a  —  d 
6a  a— d  I 

I  ffi  X  a  1 


in  which 


A.  MATHEMATICS  AND  PHYSICS. 


e\  = 


01  = 


{C  —  B  B  —  A\  1 
^  c  —  b  b  —  a)c  —  a  ’ 

1 


5  D  —  C 

C  —  B 

(  d  —  c 

c  —  b 

and  A ,  a,  I?,  b ,  C,  c,  D,  cZ,  are  known  corresponding  values  of  the 
co-ordinates  -X,  x. 

Then  will 


d  x 


*f>x  = 


c  —  b 
2 

c  —  b 


C  +  B  — 


l  +  « 


d  —  j  b  +  c 
d  —  a  [  ) 
a  —  b  -j-  c  j  ) 


_ o  r  *  d  —  *  b  -h  c 

-l  ft  d-«  \ 

3  1 


a  —  d 


2  <C+B+H }; 

in  which  H  is  substituted  for 


—j?  f  ft 


c  — J  °A  d  —  a 
3  1  .i  t~t~:  f 


Also, 


C- -  +  S**  = 


l-M 

c  —  b  r 


a  —  ^  b  —f—  c 
a  —  d 


2  b  2 

If  the  terms  be  equidistant,  that  is,  if 

d  —  c  =  c  —  b  —  b  —  a, 

H  becomes 


+  *  +  ('-=5 >)"!• 


C  +  B  —  D  +  A 

12 


Then 


dx!bX  = 


c  —  b  {  q  _ |_  £  _j_  B  —  D  A 


12 


and 
C  —  B 
2 


+s;*-V>+*+ 


The  three-point  exponential  formula 

X  =  a  -f-  £  y*  ? 

a,  /3,  and  y  being  unknown,  and  independent  of  the  variable  x,  may 
take  the  form 


MATHEMATICS. 


89 


X=A  +  (B-A)q~  1 


in  which 


«— ■— 1’ 
C  —  A 


ql~°  —  1  —  B  —  A  ’ 
whence  may  be  determined  the  value  of  q. 

B  —  A 


+  s  *  = 


dx!  X 


ii±  1 


1 


B  —  A  -\-  b  —  a  \A 


fl 


y  C-B  +  c-b)A 


B—A 

6-a__  l 


B  —  A 


2  1  ~  q  —  1 

Q  is  the  Napierian  logarithm  of  q. 

When  the  terms  are  equidistant,  i.  e.  c  —  b  =  b  —  a, 


_  (C  -  B\h~ 

-Kb  — a)  ’ 


a  .  fjL 


B  ’ 


B 


and 


B  —  A 


B  —  A 


qb~a—  1  C  —  B—B  —  A • 


fi  is  (.4342945)  the  modulus  of  the  common  system  of  logarithms. 
When  the  terms  are  not  equidistant,  the  application  of  the  expo¬ 
nential  function  involves  the  resolution  of  equations  of  higher  than  the 
first  degree. 

The  three-point  parabolic  formula 

X  =  a  +  j8  (#  —  a)y 

may  become 

X=A  +  (B-A)(^j; 

in  which 


A  .  c  — 


«  + 1 


B  —  A  .  b  —  a 


b  —  a 


8 


90 


A.  MATHEMATICS  AND  PHYSICS. 


Then  will 

*b 


xf  x  —  b— a  A  + 


B  —  A 


and 


?  +  i 


d  x 


X=c  —  bA  + 


—  a  .  C  —  A  b  —  a  .  B 


q  -\-  1  q  +  1 

Th e  finite  integral  of  (a?  —  a)q  advances  in  the  form  of  a  series ,  no 
application  of  which  has  been  made  in  the  illustrations  which  follow. 

Of  the  functions  above  enumerated,  the  algebraic  will  commonly 
prove  the  most  simple  in  practice,  but  will  not  in  all  cases  satisfy 
the  conditions  required.  When  assumed  to  express  the  law  of  relation 
between  certain  known  values  of  the  function  Lx  or  Lxvx ,  a  portion 
of  the  resulting  series  of  numbers  between  the  known  values  may 
increase  with  advancing  age,  rather  than  diminish. 

The  values  between  B  and  C,  derived  from  the  algebraic  formula 
assigning  a  law  of  relation  between  the  four  known  values  A ,  B , 
C,  and  D,  lie  between  corresponding  duplicate  values  derived  from 
two  algebraic  formulae,  one  a  function  of  the  three  known  values 
A,  B ,  and  C ,  and  the  other  of  the  three  values  B ,  C,  and  D.  When 
the  relations  of  the  known  values  to  each  other  are  such  that  the- 
series  resulting  from  each  of  the  latter  formulae  diminish  continuously 
with  advancing  age  from  A  to  C  and  from  B  to  D  respectively,  then 
that  portion  between  B  and  C  of  the  single  algebraic  series  con¬ 
necting  the  four  values  A ,  B ,  C,  and  D  must  diminish  continuously. 

The  series  of  values  resulting  from  the  algebraic  formula  assigning 
a  law  of  relation  between  the  three  known  values  A,  i?,  and  C  will 


continuously  diminish  from  A  to  C  only  when 

,  .  .  ,  ,  A  —  C 

are  each  positive  and  greater  than  _ ; 


B  and  « 


b  —  a 

or,  if  the  three  terms 


be  equidistant,  only  when  - ^  the  value  of  the  ratio  of  the  first 


differences  is  between  and  3,  and  the  differences  themselves  nega¬ 
tive.  Similar  relations  obviously  obtain  when  the  three  known  values 
are  B ,  C,  and  D. 

Applying  this  test  to  the  Prussian  Life-Table,  we  first  find  that  the 
algebraic  function  assigning  a  law  of  relation  between  the  three  known 
values  does  not  completely  satisfy  the  conditions  for  the  proportions 
surviving  ages  0,  1,  3;  3,  5,  14;  75,  85,  95;  and  85,  95,  105;  that 


MATHEMATICS. 


91 


is,  for  the  extremes  of  the  table,  the  values  there  rapidly  diminishing ; 
and  also  for  ages  3,  5,  and  14,  where  there  is  a  great  disparity  in  the 
length  of  the  intervals  of  age.  It  will  hereafter  appear  that  eccen¬ 
tricities  at  the  older  ages  may  be  disregarded  in  constructing  tables 
of  future  duration  of  life,  and  of  life-annuities,  without  materially 
affecting  the  correctness  of  the  results  for  earlier  ages. 

TABLE  I. 

Average  Future  Duration  op  Life  in  Prussia. 

Algebraic  Integration. 


1  ^ 

Proportions  Born,  and 
Living  at  Specified  Ages 
in  Prussia,  calculated,  by 
the  Integral  Method, 
from  three  Consecutive 
Ratios  of  Deaths  to 
Population. 

Sum  of  ( Lx ) 
the  Propor¬ 
tions  Living 
at  the  given 
Age  and  at 
all  subse¬ 
quent  Ages 
specified.  | 

1 

Aggregate  Number  of  Future  Years 
that  (Lx)  the  Proportions 
surviving  Specified  Ages  will  live. 

Average 
Future  1 
Duration  of 1 
Life. 

£*+£*+10 
+  £*+20 

+  •••• 

10$  ,  8*  +  8*  +  n 

"2)  ,  ^*  +  ^*+10  ^*—10  ^-20 

\ 

£*  ! 

<  +  ■"  12 

So 

Lx 

Ages. 

Lx 

dx  r  l 

J  X  x 

Ex 

5 

69,916 

5 

69,916 

364,916 

!l4 

64,249 

15 

63,748 

295,000 

2,626,778 

41.21 

!  25 

59,159 

25 

59,159 

231,252 

2,012,408 

34.02  ! 

35 

53,386 

35 

53,386 

172,093 

1,448,721 

27.14  1 

45 

46,488 

45 

46,488 

118,707 

948,047 

20.39  | 

55 

37,585 

55 

37,585 

72,219 

524,773 

13.96 

65 

23,706 

65 

23,706 

34,634 

215,941 

9.11  1 

75 

9,104.2 

75 

9,104.2 

10,927.6 

54,597 

6.00  : 

85 

1,726.7 

85 

1,726.7 

1,823.4 

5,847 

3.39 

95 

96.1 

96.7 

—  232 

105 

.64 

.64 

In  the  preceding  table  the  first  of  the  columns  headed  Lx  gives  the 
proportions  surviving  certain  ages  according  to  the  law  of  mortality 
prevailing  in  Prussia ;  the  probability  of  surviving  each  of  the  several 
intervals  being  calculated  from  three  consecutive  ratios  of  deaths  to 
populations.  In  the  second  of  the  columns  headed  Lx  the  values  for 
ages  95  and  105  were  computed  from  the  logarithms  of  the  propor¬ 
tions  surviving  ages  65,  75,  and  85,  by  the  exponential  formula 
which  expresses  the  value  of  the  required  logarithms  in  terms  of  the 
age,  and  of  the  three  given  logarithms.  The  number  surviving  age 
15  (63,748)  was  obtained  by  assuming  an  algebraic  law  of  relation 
for  the  proportions  surviving  the  four  ages  5,  14,  25,  and  35. 


92 


A.  MATHEMATICS  AND  PHYSICS. 


The  next  column  ( Sx )  gives  the  sum  of  the  proportions  surviving 
the  given  age  and  all  subsequent  specified  ages. 

z*105 

The  values  in  column  headed  d  x  J  -  Lx  give  the  aggregate 

number  of  future  years  of  life  that  the  proportions  surviving  given 
ages  will  enjoy,  according  to  the  prevailing  law  of  mortality,  and 
were  each  computed  from  four  equidistant  values  in  the  preceding 
column  ( Sx ),  by  means  of  the  formula 


^•*105  „  ( 

Lx  =  -  }S.+  &+,  + 


Sx  +  sx 


12 


The  values  thus  obtained,  divided  by  the  corresponding  proportions 
living,  give  the  average  future  duration  of  life. 

We  have  already  called  attention  to  the  unsatisfactory  nature  of 
the  values  resulting  from  the  use  of  the  algebraic  formula  when  the 
given  numbers  rapidly  diminish,  as  in  the  Life-Table  after  about 
age  75. 


Table  II.  will  compare  corresponding  results  obtained  by  different 
1  formulae  and  processes. 

In  the  third  column  of  Table  II.  the  integrations  were  effected 
by  the  exponential  formula  when  the  three  given  values  involved  in 
the  equation  were  equidistant ;  when  not  equidistant,  the  parabolic 
formula  was  adopted. 

The  parabolic  and  the  exponential  formulae  each  afford  results  that 
constantly  diminish  with  advancing  age. 

The  process  of  integration  by  the  algebraic  formula  involving  four 
known  values  is  the  simplest,  and  between  ages  15  to  75  is  entirely 
satisfactory ;  from  5  to  15,  and  from  75  upwards,  the  values  afforded 
are  not  so  reliable  ;  and  from  95  to  105,  duration  of  life  is  represented 
as  negative. 


In  Table  IV.  we  observe  that  the  first  three  columns  of  average 
future  duration  of  life  present  results,  for  the  larger  part  of  life, 
almost  identical. 

Values  by  the  algebraic  formula  slightly  exceed  those  of  the 
following  column,  calculated  by  a  combination  of  parabolic,  expo¬ 
nential,  and  algebraic  formulae.  The  excess  at  specified  ages  from 
15  to  55  inclusive  is  only  the  one-hundredth  part  (.01)  of  a  year,  or 
about  four  days. 


MATHEMATICS. 


93 


TABLE  II. 

Comparison  op  Temporary  Aggregate  Future  Duration  of  Life, 

CALCULATED  BY  DIFFERENT  METHODS,  FROM  THE  PROPORTIONS  SURVIVING 
ACCORDING  TO  THE  PRUSSIAN  LiFE-TABLE. 


Ages. 

Proportions 
Born  alive, 
and  Surviving 
certain  Ages. 

The  Aggregate  Number  of  Years  of  Life  which  the  Proportions 
Surviving  at  the  Commencement  of  certain  Intervals  of  Age  will  enjoy 
during  each  Interval. 

-^x+n  Ar 

^x  +  n  +  4: 

C 

ix  r+" 

2 

X 

Lx 

J  X  * 

+  ?Tlx 

2 

Parabolic  and  Exponential 

Algebraic 

By  Annual 

Equidifferent 

Method. 

Duplicates. 

Mean. 

Formula. 

Interpolation. 

0 

1 

100,389 

82,941 

87,827* 

155,560* 

155,176 

91,665 

1’55,494 

91,665  ! 
156,578 

3 

73,637 

142,992 

142,359* 

142,792 

143,251 

143,553 

5 

69,916 

664,403*  j 
666,802  l 

f 

665,602 

656,124 

661,937 

668,320  ; 

15 

63,748 

613,405  | 

615,410  1 

614,407 

614,370 

616,016 

614,535 

25 

59,159 

563,826  1 

563,581  l 

f 

563,653 

563,687 

563,655 

562,725 

35 

53,386 

500,393  ] 

500,836  j 

I 

500,614 

500,674 

500,322 

499,370 

45 

46,488 

422,257  ] 

423,648  j 

422,952 

423,274 

422,990 

420,365 

55  | 

37,585 

311,573  j 
306,973  j 

1 

309,273 

308,830 

311,376 

306,455 

65 

23,706 

164,599  } 

155,807  j 

160,203 

161,341 

159,662 

164,050 

75 

9,104 

49,992  ] 
45,211  j 

1 

47,601 

48,750 

47,769 

54,155 

85 

1,727 

7,137  j 
5,688  j 

6,412 

6,081 

6,368 

9,115  ; 

95 

96 

285  j 
209  j 

247 

—  194 

228 

485 

105 

1 

115 

0 

1 

A  comparison  of  the  results  in  the  third  column  of  Table  IV.  with 
those  arrived  at  by  a  general  interpolation  and  direct  summation  of 
the  proportions  surviving  each  anniversary  of  birth,  exhibits  a  differ- 


*  These  four  values  were  calculated  by  the  parabolic  formula ;  the  other  values 
in  the  column  by  the  exponential. 


94 


A.  MATHEMATICS  AND  PHYSICS. 


ence  at  specified  ages  from  birth  to  age  85  inclusive,  that  in  but  one 
case  (at  age  65)  exceeds  three  one-hundredth  parts  (.03)  of  one  year, 
or  about  eleven  days. 

The  former  results  are  deemed  in  every  respect  as  satisfactory  as 
the  latter. 

We  observe  that  results  by  the  equidifferent  method  compared  with 
approved  results,  from  birth  to  age  45  inclusive,  are.  usually  about 
one  tenth  of  a  year  in  excess  ;  and  that  for  ages  above  45  the  excess 
is  much  greater. 


TABLE  III. 

Euture  Duration  of  Live  in  Prussia. 

The  Temporary  Future  Duration  of  Life  for  the  Proportions  Surviving,  was  computed  hy 
the  Parabolic  Formula  from  Birth  to  Age  1  ;  Exponential,  from  1  to  3  and  3  to  5  ; 
Mean  of  Parabolic  and  Exponential,  from  5  to  15  ;  Algebraic,  from  15  to  75  ;  and 
Mean  of  Exponential  Duplicates,  from  75  to  105. 


• 

Ages. 

Temporary  Future 
Duration  of  Life. 

Future 

Duration  of  Life. 

Average  Future 
Duration  of  Life. 

X 

d*f:+nL* 

dx  r05  l 

J  X  X 

dx  rmLx 

0 

87,827 

3,678,033 

36.64 

1 

155,176 

3,590,206 

43.29 

3 

142,992 

3,435,030 

46.66 

5 

665,602 

3,292,038 

47.09 

15 

614,370 

2,626,436 

41.20 

25 

563,687 

2,012,066 

34.01 

35 

500,674 

1,448,379 

27.13 

45 

423,274 

947,705 

20.39 

55 

308,830 

524,431 

13.95 

65 

161,341 

215,601 

9.09 

75 

47,601 

54,260 

5.96 

85 

6,412 

6,659 

3.85 

95 

247 

247 

2.57 

MATHEMATICS, 


95 


TABLE  IV. 

Comparison  of  Average  Future  Duration  of  Life. 
Computed ,  by  different  Processes,  from  the  Prussian  Life-Table. 


Ages. 

By  the 
Algebraic 
Formula. 

Parabolic,  0  —  1 ; 
Exponential,  1-3  and  3  -  5  ; 

Mean  of  Parabolic  and  Exponential,  5-15; 
Algebraic,  15  -  75 ; 

Mean  of  Exponential  Duplicates, 75  - 105. 

By  Annual 
Interpolation. 

Assuming  Lx 
within  each 
Interval  to  advance 
by  an 

Equidifferent 

Progression. 

0 

36.64 

36.66 

36.77 

1 

43.29 

43.27 

43.40 

3 

46.66 

46.63 

46.76 

5 

47.09 

47.06 

47.19 

15 

41.21 

41.20 

41.17 

41.28 

25 

34.02 

34.01 

34.02 

34.09 

35 

27.14 

27.13 

27.14 

27.24 

45 

20.39 

20  39 

20.40 

20.53 

55 

13.96 

13.95 

13.98 

14.21 

65 

9.11 

9.09 

9.03 

9.61 

75 

6.00 

5.96 

5.97 

7.00 

85 

3.39 

3.85 

3.82 

5.55 

95 

2.57 

2.37 

5.05 

TABLE  V. 

Value,  at  certain  Ages,  of  One  Dollar  to  be  paid  at  the  End  of 
each  Year  during  the  Remainder  of  Life,  according  to  the  Prus¬ 
sian  Life-Table,  with  Process  for  determining. 

Interest  of  Money ,  Four  per  Cent.  Algebraic  Finite  Integration. 


I  Ages. 

L  (-L)* 

L'x  +  ^  +  10  + 

—  \Sf  +  £'  ,in-f 

2  *  *  *+10  100  * 

•§T  L’x 

*  Vl.04/ 

£'*  +  20  “H  •  •  •  • 

£'* 1 

X 

£'* 

—  ^  +  ^,05  L< 

2  1  x  x 

Life- Annuity, 

4  per  Cent. 

1'  5 

57,466 

143,287 

i  15 

35.397 

85,821 

666,675 

18.33 

j  25 

22,192 

50,424 

384,260 

16.82 

j  35 

13,529 

28,232 

208,804 

14.93 

i  45 

7,959 

14,703 

103,448 

12.50 

55 

4,347 

6,744 

43,186 

9.44 

65 

1,852 

2,397 

13,115 

6.58 

75 

481 

545 

2,306 

4.29 

85 

61.6 

63.9 

133.6 

1.67 

95 

2.3 

2.3 

—  13.8 

105 

.01 

.01 

ff.-lO  S'x— so¬ 


il  —  S'x  -}-  S'z-fio 


96 


A.  MATHEMATICS  AND  PHYSICS. 


In  Table  V.  we  observe  that  the  ratios  of  the  first  differences  of  the 

values  ( Lx  second  column,  for  ages  65  and  over,  are 

not  within  the  required  limits,  J  and  3,  and,  consequently,  that  the 
values  of  the  annuity  resulting  from  integration  by  the  algebraic  for¬ 
mulae  are  to  an  extent  unsatisfactory. 

Had  the  integration  of  S'x  for  the  older  ages  been  effected  by  the 
exponential  formula,  the  following  would  have  resulted. 


1  . 

1 

!  Ages. 

-^  +  ST&\ 

sr  si. 

Duplicates. 

Mean. 

-1 

55 

j  43,539.1  ) 

(  42,671.2  J 

43,105 

9.42 

65 

J  13,417.5  ) 

|  12,711.6  J 

13,065 

\ 

6.55 

75 

j  2,525.4  ) 

\  2,281.3  \ 

2,403 

4.50 

85 

1 

(  233.3  \ 

]  188.2  J 

210.8 

* 

2.92 

In  Table  VI.  the  values  in  the  third  column  corresponding  to  inter¬ 
vals  between  ages  65  and  95  are  arithmetical  means  of  dupli¬ 
cate  values  resulting  from  finite  integration  of  L'x  by  the  exponential 
formula.  The  value  opposite  age  95  is  a  single'  value  similarly  ob¬ 
tained. 


Duplicate  Values  and  Mean. 


Ages. 

Duplicates. 

Mean. 

65 

j  10,990.8  i 
1 10,356.0  ■ 

|  10,673 

75 

f  2,312.6  i 
1  2,076.1  i 

[  2,194 

85 

f  229.4  ( 

1  183.0  * 

Single  Value. 

■  206.2 

95 

6.3 

6.3 

MATHEMATICS. 


97 


In  the  same  column  (the  third)  the  value  of  the  summation  from  5 
to  15  (455,694)  is  the  arithmetical  mean  of  a  result  (456,371)  derived 
by  integration  from  the  parabolic  formula  involving  values  in  the 
preceding  column  for  ages  3,  5,  and  15,  and  of  another  (455,017) 
obtained  by  finite  integration  from  the  exponential  formula  involving 
values  for  ages  5,  15,  and  25. 

The  remaining  values  in  that  column  were  obtained  by  the  finite 
integration  of  a  series  of  distinct  algebraic  formulae,  each  involving 


four  given  values  of  Lx 


When  the  terms  in  the  second  column  ( L'x )  are  equidistant,  and 
the  intervals  unity,  1  —  <rh)  in  the  expression  for  the  finite 


integral  of  the  algebraic  formula  becomes  zero ,  and  the  correspond- 

L'  4-  U 

ing  values  in  column  third  become  — — . 


In  the  second  column  (X2^2  and  T4v4)  values  for  ages  2  and  4  were 
interpolated  by  means  of  the  exponential  formula  involving  values  for 
ages  1,  3,  and  5. 

The  value  of  the  life-annuity  ( ax )  at  each  of  the  specified  ages  was 
obtained  by  dividing  the  corresponding  value  in  the  fourth  column  by 
that  in  the  second,  and  from  the  quotient  deducting  five  tenths  of 
unity. 

We  observe  that  annuities  from  age  15  to  45  inclusive,  according 
to  Table  V.,  and  for  subsequent  specified  ages  according  to  the 
modification  of  that  table  by  the  exponential  formula,  are  essentially 
identical  with  values  in  Table  VI.  at  corresponding  ages. 

Our  L'x  (constructed  according  to  Barrett’s  method)  corresponds  to 
the  Dx  of  Mr.  Griffith  Davies  and  later  writers.  Our  '^j*06  L'x  (or  * 
$*)  ls  the  Nx  employed  by  Dr.  Farr  and  Mr.  Gray,  and  the 
Nx_l  of  Mr.  Davies,  adopted  by  Mr.  David  Jones,  Mr.  Jenkin  Jones, 
Professor  De  Morgan,  and  others. 

Unaugmented  annual  and  single  premiums  to  insure  $  100,  payable 
at  the  end  of  the  year  of  decease,  may  be  computed  by  the  formulae 
commonly  employed,  and  heading  the  respective  columns,  or  be  taken 
directly  from  Mr.  Orchard’s  very  useful  tables  of  “Assurance  Pre¬ 
miums.” 


9 


98 


A.  MATHEMATICS  AND  PHYSICS. 


TABLE  VI. 

Life-Annuity,  with  Tables  preparatory  ;  also  Tables  of  Unaug¬ 
mented  Annual  and  Single  Premiums  to  insure  $  100,  payable  at 
the  End  of  the  Year  in  which  Life  shall  terminate. 

Interest  of  Money ,  Four  per  Cent  per  Annum.  Integration  by  different  Formulae. 


Ages. 

L  (—\X 
x  VI. 04  J 

0 

w 

+ 

i 

Life  Annuities, 
4  per  Cent. 

Premiums 

Unaugmented. 

Annual. 

Single. 

+ 

-t 

fi 

1 

e 

+ 

r 

<N 

to 

w* 

H 

/- 

^  1 
© 

s. 

'C 

c 

■n 

© 

r 

H 

53 

+ 

✓ 

5 

n 

c 

53 

+ 

5 1*3 

r 

X 

© 

o 

0 

100,389 

90,070 

1,478,812 

14.23 

2.72 

41.42 

1 

79,751 

75,565 

1,388,743 

16.91 

1.74 

31.12 

2 

71,380 

68,422 

1,313,178 

17.90 

1.45 

27.31 

3 

65,463 

63,236 

1,244,746 

18.51 

1.28 

24.96 

4 

61,010 

59,238 

1,181,520 

18.87 

1.19 

23.58 

5 

57,466 

455,694 

1,122,282 

19.03 

1.15 

22.96 

15 

35,397 

282,414 

666,588 

18.33 

1.33 

25.65 

25 

22,192 

175,456 

384,174 

16.82 

1.77 

31.46 

35 

13,529 

105,356 

208,718 

14.93 

2.43 

38.73 

45 

7,959 

60,261 

103,362 

12.49 

3.57 

48.12 

55 

4,347 

30,021 

43,101 

9.42 

5.75, 

59.92 

65 

1,852 

10,673 

13,080 

6.56 

9.38 

70.92 

75 

481 

2,194 

2,407 

4.50 

14.34 

78.85 

85 

61.6 

206.2 

212.5 

2.95 

21.47 

84.81 

95 

2.3 

6.3 

6.3 

105 

.0, 

Methods  for  determining  from  the  above  data  the  values  of  other 
single  life  benefits,  whether  uniform,  increasing,  or  decreasing,  either 
for  the  entire  period  of  life  or  for  limited  portions,  may  readily  be 
devised. 

So  also  methods  analogous  to  those  employed  in  framing  the  pre¬ 
ceding  tables  may,  with  advantage,  be  adopted  in  constructing  tables 
that  shall  afford  facilities  for  the  ready  solution  of  questions  involving 
two  or  more  life  contingencies. 

We  now  wish  rules  for  determining,  by  brief  processes,  values  in¬ 
termediate  between  those  already  obtained. 

Either  of  the  formulae  already  given  may  be  resorted  to  ;  of  which 
the  following  is  the  simplest. 


MATHEMATICS. 


99 


X=  A 
+  B 
+  C 


x  —  b  .  x  —  c 


a  —  b 

.a  —  c 

x  —  a 

.  x  —  c 

b  —  a 

.  b  —  c 

x  —  a 

.  x  —  b 

c  —  a 

.c  —  b 

A ,  a,  j E?,  5,  and  C,  c  being  known  corresponding  values  of  the  co¬ 
ordinates  X  and  x. 

The  algebraic  formula  involving  four  given  values  will  commonly 
afford  results  of  a  nature  entirely  satisfactory  within  the  usual  limits 
of  inquiry. 

Given,  A,  a,  B,  Z>,  C,  c,  and  D,  d,  corresponding  known  values  of 
X ,  x ;  required  values  intermediate  between  B  and  C.  If  the  given 
terms  be  equidistant,  and 

n—d  —  c  —  c  —  b  =  b  —  a, 
the  algebraic  formula  will  give  the  following  :  — 


TABLE  VII. 

Special  Formulae  for  Interpolation,  involving  Four  known  Equi¬ 
distant  Values  of  the  Function. 


Algebraic. 


Ages. 

X 

*  + 

Tff  n 

1 

Tff 

(9 

B+  C) 

+ 

3 

Sffffff 

(3 

.  9  B 

+ 

C- 

-19  A- 

-11  D) 

+ 

T20  n 

1 

Tff 

(8 

5  +  2  C) 

+ 

8 

Tffffff 

( 

8  B 

+  2 

c- 

-  6  A- 

-4  D) 

+ 

Iff  71 

1 

Iff 

(7 

5  +  3  C) 

+ 

affffff 

(3 

.IB 

+  3 

C- 

-  17^4  — 

-  13  D) 

+ 

Tff  W 

1 

Tff 

(6 

5  +  4  C) 

+ 

4 

Tffffff 

(3 

.65 

+  4 

C- 

-  16  A  - 

-  14  D) 

+ 

Iff  n 

1 

? 

( 

B+  C) 

+ 

Tff 

( 

B 

+ 

C- 

-  A- 

-  D) 

+ 

16ff  n 

1 

1  ff 

(4 

5  +  6  C) 

+ 

4 

Tffffff 

(3 

.4  B 

+  6 

C- 

-UA- 

-16D) 

+ 

Tff  71 

1 

TO 

(3 

5  +  7  C) 

+ 

tffffff 

(3 

.3  B 

+  7 

C- 

-  13  A- 

-17  D) 

+ 

Tff  71 

1 

Tff 

(2 

5  +  8  C) 

+ 

8 

Tffffff 

( 

2  B 

+  8 

C- 

-  4A- 

-  6  D) 

»  + 

Tff  n 

Tff 

( 

5  +  9  C) 

+ 

affffff 

(3 

.  B 

+  9 

c- 

-11  A- 

-19  D) 

Example. —  Given,  unaugmented  annual  premiums,  from  Table 
VI.,  corresponding  to  ages  15,  25,  35,  and  45  ;  required  the  premium 
for  age  28. 


100 


A.  MATHEMATICS  AND  PHYSICS. 


The  difference  between  ages  25  and  28  is  of  (n)  the  interval  of 
age  from  25  to  35  ;  that  is  28  is  b  “h  t3o  n- 
Then 

-^28  —  tV  B  ~h  ^  C)  -|-  (3 . 7  B  -f-  3  C  —  17  A  —  13  D), 

in  which  A,  B ,  C,  and  D  equal  respectively  1.33,  1.77,  2.43,  and 
3.57. 

TV(7JB  +  3C)  =  1.968 
3  (7  B  +  3  C)  ==  59.04 
11A+13B  =  69.02; 

.-.  V28  =  1.93. 

Required  a  value  corresponding  to  age  40,  from  data  in  the  column 

headed - -  -j-  JJ x  .  The  formula  is 

£ 

X»  =  i  (5  +  C)  +  TV  (5  +  C  -  A  -  D), 

and  the  given  values  are  for  ages  25,  35,  45,  and  55. 

B  +  C  =  312,080, 

A  -f  D  =  427,275  ; 

Xi0  =  148,840. 

Table  VII.  may  take  the  symmetrical  form  of 
TABLE  VIII. 


Special  Formula:  for  Interpolation,  involving  Four  known  Equi¬ 
distant  Values  op  the  Function. 

Algebraic. 


Ages. 

X 

X 

b  "h  io n 

A  (9  C) 

9x1(3.95+  C—  19  A—  11  D) 

4(85  +  2  0) 

8x2  (3. 85  +  20—  18  4  —  12  5) 

4  (7  -B  +  3  C) 

7x3(3.75  +  30—  17  A—  13  5) 

4(65  +  4C) 

6x4(3.65  +  4  0—  16  A  — 14  5) 

is  n 

4(55  +  5  0)  +  ^ 

5x5(3.5  5  +  5  0—  15  4—  15  D)j 

A» 

5*5(4£  +  6C) 

4x6  (3. 45  +  60  —  14  —  16  5) 

t7o  n 

4(3  5  +  7  0) 

3x7  (3. 35  +  70  —  13  A  — 17  5) 

4(25  +  8  0) 

2x8  (3. 25  +  80 —  12  j!  —  18  5) 

^+I0W 

4  (  5  +  9  C) 

1x9(3.  5  +  9C  — 11 A  — 19  5) 

v  / 

ASTRONOMY. 


101 


And  generally,  when  the  four  terms  A,  B,  C ,  and  D  are  equidistant, 
*  =  c -I  >  —  n+x~ic)  +  6(c-^  X 

jc  — x .  x  —  &  [3  .  (c  —  x  B  -\-x  —  b  C)  —  d  — x  A  —  x  —  a  D~\\. 

The  writer  is  not  aware  that  any  previous  attempt  has  been  made 
to  pass  by  direct  and  summary  processes  from  the  immediate  results 
of  actual  observations  to  solutions  of  monetary  and  other  practical 
questions  involving  life  contingencies,  as  accurate  and  reliable  as 
those  obtained  by  the  intervention  of  a  formidable  interpolation.  In 
view  of  the  large  and  rapidly  accumulating  mass  of  population  and 
mortality  statistics,  such  processes  seem  to  be  demanded. 

The  approximate  methods  heretofore  published  have  already  been 
adverted  to  ;  allusion  has  also  been  made  to  certain  formulae  adopted 
in  the  construction  of  theoretical  Life-Tables,  which  afford  facilities 
for  the  independent  formation  of  required  monetary  and  other  values. 

Note.  —  The  average  future  duration  of  life  for  ages  15,  25, 35, 45, 
and  55,  deduced  from  values  in  column  C,  on  page  82,  are  from  .1  to 
.2  of  a  year  less ,  and  those  derived  from  values  in  column  E  are  from 
.2  to  .3  of  a  year  greater ,  than  corresponding  durations  obtained  by 
more  accurate  methods.  Arithmetical  means  of  these  results  exceed 
the  true  values  by  about  .05  of  a  year.  By  giving  greater  compara¬ 
tive  weight  to  values  deduced  from  C,  closer  approximations  will 


ensue. 


